+
+ |
+ 著者
+ |
+
+ Toyoda Masahiro, Kogiso Nbzomu
+ |
+
+
+ |
+ journal or publication title
+ |
+
+ Journal of Mechanical Science and Technology
+ |
+
+
+ |
+ volume
+ |
+
+ 29
+ |
+
+
+ |
+ number
+ |
+
+ 4
+ |
+
+
+ |
+ page range
+ |
+
+ 1361-1367
+ |
+
+
+ |
+ year
+ |
+
+ 2015-04-14
+ |
+
+
+ |
+ 権利
+ |
+
+ The final publication is available at link.springer.com via http://dx.doi.org/10.1007/s12206-015-0305-9.
+ |
+
+
+ |
+ URL
+ |
+
+ http://hdl.handle.net/10466/15650
+ |
+
+
+
+doi: 10.1007/s12206-015-0305-9
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diff --git a/samples/texts/2926839/page_7.md b/samples/texts/2926839/page_7.md
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+Fig. 11. Cross-sectional area distributions
+
+usefulness of this method was demonstrated.
+
+* An accurate Pareto set is obtained by parametrically changing the aspiration level, because each Pareto solution is obtained using a mathematical programming method.
+* It was shown that the proposed method could be used to investigate the effect of the variation of random variables on the shape of the Pareto frontier. In addition, the shift of each Pareto solution with the same aspiration level could be traced with respect to the variation of uncertain parameters.
+* This method makes it possible to investigate how the variation of the design variables and parameters affect the Pareto set. This investigation is possible without obtaining full Pareto set.
+
+## Acknowledgment
+
+Part of this research was supported by JSPS KAKENHI 26249131. The authors wish to express their appreciation to
+
+Prof. S. Kitayama at Kanazawa University for his valuable comments on this research.
+
+## References
+
+[1] G. Park, T. Lee, K. Lee, and K. Hwang, Robust Design: An Overview, *AIAA Journal*, 44(1) (2006) 181-191.
+[2] H. Beyer and B. Sendhoff, Robust Optimization–A Comprehensive Survey, *Computer Methods in Applied Mechanics and Engineering*, 196(33-34) (2007) 3190-3218.
+[3] K. M. Mittinen, *Nonlinear Multiobjective Optimization*, Kluwer Academic Publishers (2004).
+[4] H. Nakayama and Y. Sawaragi, Satisficing Trade-Off Method for Multiobjective Programming, *Lecture Notes in Economics and Mathematical Systems*, 229 (1984) 113-122.
+[5] H. Nakayama, K. Kaneshige, S. Takemoto, and Y. Watada, An Application of a Multiobjective Programming Technique to Construction Accuracy Control of Cable Stayed Bridges, *European. Journal of Operational Research*, 83(3) (1995) 731-738.
+[6] S. Kitayama and K. Yamazaki, Compromise Point Incorporating Trade-off Ratio in Multiobjective Optimization, *Applied Soft Computing*, 12(8) (2012) 1959-1964.
+[7] H. Nakayama, Trade-off Analysis Using Parametric Optimization Techniques, *European. Journal of Operational Research*, 60(1) (1992) 87-98.
+[8] A. H-S. Ang and W. H. Tang, *Probabilistic Concepts in Engineering Planning and Design, Vol. 1, Basic Principles*, John Wiley & Sons (1975).
+[9] W. Chen, M. M. Wiecek, and J. Zhang, Quality Utility - A Compromise Programming Approach to Robust Design, *Journal of Mechanical Design*, 121(2) (1999) 179-187.
+[10] R. T. Haftka and Z. Gürdal, *Elements of Structural Optimization*, Kluwer Academic Publishers (1992).
+
+Masahiro Toyoda is presently M.S. candidate in Department of Aerospace Engineering at Osaka Prefecture University in Japan. He received his B.S. degree in aerospace engineering from Osaka Prefecture University, Japan in 2012. His research interest is multiobjective optimization and its application to aerospace structural systems.
+
+Nozomu Kogiso is presently an Associate Professor in Department of Aerospace Engineering at Osaka Prefecture University in Japan. He received his B.S. degree in physics from Nagoya University, Japan in 1988 and his M.S. and Dr. Eng. from Osaka Prefecture University in 1994 and 1997, respectively. His research interests include robust and reliability-based design optimization.
\ No newline at end of file
diff --git a/samples/texts/2952501/page_30.md b/samples/texts/2952501/page_30.md
new file mode 100644
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+# 7 Distances along the half-plane boundary
+
+To fulfill our goal of showing the asymptotic equivalence between the oriented and non-oriented distances in uniform Eulerian triangulations, we need as a technical ingredient some estimates on the (oriented) distances along the boundary of $\mathcal{L}$.
+
+Note that the vertices on $\partial\mathcal{L}$ are of two types, those of coordinates $(i, 0)$ for some $i \in \mathbb{Z}$, and those of coordinates $(i + 1/2, \varepsilon)$, for some $i \in \mathbb{Z}$. To simplify notation, the results in this section only deal with the distances between vertices of the first type, since we are interested in asymptotic estimates, and including the vertices of the second type only adds 1 or 2 to the considered distances. We will lay the stress on this generalization whenever it arises later in the paper.
+
+In the sequel, we will use leftmost mirror geodesics, that were defined in Section 5 for finite cylinder triangulations, and that we generalize now to $\mathcal{L}$. For any $i \in \mathbb{Z}$, the leftmost mirror geodesic from $(i, 0)$ in $\mathcal{L}$ is an infinite path $\omega$ in $\mathcal{L}$, whose reverse is an oriented geodesic, and that visits a vertex $\omega(n)$ in $\mathcal{L}_n$ at every step $n \ge 0$. It starts at $(i, 0)$, and is obtained by choosing at step $n+1$ the leftmost edge between $\omega(n)$ and $\mathcal{L}_{n+1}$. As before, for $i < j$, the leftmost mirror geodesics from $(i, 0)$ and $(j, 0)$ will coalesce before hitting $\mathcal{L}_r$, if and only if all the trees $\mathcal{T}_i, \mathcal{T}_{i+1}, \dots, \mathcal{T}_{j-1}$ all have height strictly smaller than $r$.
+
+## 7.1 Block decomposition and lower bounds
+
+We first want to obtain lower bounds on the distances along the boundary of $\mathcal{L}$. For that purpose, we adapt the **block decomposition** of causal triangulations [14, Section 2.1], to $\mathcal{L}$.
+
+Figure 16: The block of height 3 between $\mathcal{T}_0$ and $\mathcal{T}_1$ in the triangulation of Figure 15. As before, ghost modules are shown in pale grey.
+
+For $r \ge 1$, we define the random map $G_r$ to be the planar map obtained from $\mathcal{L}_{[0,r]}$ by keeping only the faces and edges that are between $\mathcal{T}_0$ and $\mathcal{T}_{i_r}$, where $i_r$ is the smallest integer $i > 0$ such that $\mathcal{T}_i$ has height at least $r$. More precisely, we only keep the skeleton modules that are at height smaller than or equal to $r$, belonging to trees $\mathcal{T}_i$, with $0 \le i \le i_r$, and the slots that are to the left of all these skeleton modules (see Figure 16 for an example). Thus, $G_r$ has one boundary that is naturally divided into four parts: the upper and lower parts that it shares with $\mathcal{L}_{[0,r]}$, and the left and right parts.
+
+Note that $\mathcal{L}$ contains a lot of submaps that have the same law as $G_r$: if $\mathcal{T}_i, \mathcal{T}_j$ are two
\ No newline at end of file
diff --git a/samples/texts/2952501/page_31.md b/samples/texts/2952501/page_31.md
new file mode 100644
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@@ -0,0 +1,37 @@
+consecutive trees reaching height $r$ in the skeleton of $\mathcal{L}$ (with $i < j$), we can define the submap of $\mathcal{L}_{[0,r]}$ encased between $\mathcal{T}_i$ (strictly) and $\mathcal{T}_j$ (included), which is obtained by keeping only the skeleton modules belonging to trees $\mathcal{T}_k$, with $i < k \le j$, and the slots that are to the left of all these skeleton modules. Such a map has the same law as $\mathcal{G}_r$.
+
+We call any map that can be a realization of $\mathcal{G}_r$, a **block of height r**.
+
+We define the **diameter** of $\mathcal{G}_r$, denoted $Diam(\mathcal{G}_r)$ to be the minimal oriented distance from a vertex on its left boundary, to a vertex on its right boundary. Note that this diameter is not uniformly large when $r$ is large. However, we will now show that a long block is also typically wide. To do so, we consider the **median diameter** of a block.
+
+**Definition 7.1.** For any $r \ge 1$, let $f(r)$ be the median diameter of $\mathcal{G}_r$, that is, the largest number such that
+
+$$\mathrm{P}(\mathrm{Diam}(\mathcal{G}_r) \ge f(r)) \ge \frac{1}{2}.$$
+
+We show the following upper bound on the median diameter, which is similar to the first part of [14, Theorem 5]:
+
+**Theorem 7.2.** There exists $c > 0$ such that
+
+$$f(r) \ge cr,$$
+
+for all $r$ sufficiently large.
+
+Let us introduce a bit of notation before explaining how to prove this theorem. For any $m \ge 1$ and $h \ge 0$, consider the layer $\mathcal{L}_{[h,h+m]}$: it is composed of a (bi-infinite) sequence of blocks of height $m$, $(\mathcal{G}_m(i,h))_{i \in \mathbb{Z}}$. To avoid ambiguities, we set $\mathcal{G}_m(1,h)$ to be the block that has a part of $\mathcal{T}_k$ as its right boundary, where $k$ is the smallest integer $l \ge 1$ such that $\mathcal{T}_l$ has height at least $h+m$. For fixed $h,m$, these blocks are independent and distributed as $\mathcal{G}_m$. For $r \ge h+m$, we denote by $N_r(m,h)$ the maximal index $i$ such that the block $\mathcal{G}_m(i,h)$ is a sub-block of $\mathcal{G}_r$. (Note that, by our convention, the minimal such index is $i=1$, so that $N_r(m,h)$ is also the number of blocks $\mathcal{G}_m(i,h)$ that are sub-block of $\mathcal{G}_r$.)
+
+To prove Theorem 7.2, we use, like in [14], a renormalization scheme, splitting $\mathcal{G}_r$ into smaller blocks. This relies on an estimate of the numbers $N_r(2m, lm)$:
+
+**Lemma 7.3.** There exists $c > 0$ such that, for every $1 \le m \le cr$, we have
+
+$$\mathrm{P}\left(\inf_{0 \le l \le (r/m)-2} N_r(2m, lm) \ge c \left(\frac{r}{m}\right)^2\right) \ge \frac{7}{8}.$$
+
+The proof of this lemma can be adapted straightforwardly from the proof of [14, Lemma 1], in the case $\beta = 2$.
+
+**Proposition 7.4.** There exists $C > 0$ such that, for any integer $m$ with $1 \le m \le Cr$, we have
+
+$$f(r) \ge C \cdot \min\left\{m, \left(\frac{r}{m}\right)^2 f(m)\right\}.$$
+
+*Proof.* Let us give a sketch of the proof of this result, as it is very similar to the one of [14, Proposition 1]. The idea is to consider the shortest (oriented) path going from a vertex on the left boundary of $\mathcal{G}_r$, to its right boundary, and whether or not it leaves a small horizontal layer of a specific type.
+
+More precisely, we pick a vertex $x$ on the left boundary of $\mathcal{G}_r$, at a height $0 \le j \le r$. Then, we can find an integer $l$ such that $x$ is located in the layer $\mathcal{L}_{[lm,(l+2)m]}$, with $0 \le l \le (r/m)-2$ and so that $|lm-j| \ge m/3$ and $|(l+2)m-j| \ge m/3$. Consider then the shortest oriented path from $x$ to the right boundary of $\mathcal{G}_r$. Either it stays in that layer, or it leaves it at some point.
+
+If it leaves the layer, then its length is bounded below by $m/3$.
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diff --git a/samples/texts/2952501/page_32.md b/samples/texts/2952501/page_32.md
new file mode 100644
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@@ -0,0 +1,69 @@
+Convergence of Eulerian triangulations
+
+If it does not leave this layer, then its length is bounded below by
+
+$$
+\sum_{i=1}^{N_r(2m,lm)} \mathrm{Diam}(\mathcal{G}_{2m}(i,lm)).
+$$
+
+Then, from Lemma 7.3, and from the definition of f, we get that, for r/m large enough,
+for some c' > 0 independent of r, l, m,
+
+$$
+P\left(\{N_r(2m, lm) < c \left(\frac{r}{m}\right)^2\} \cup \{\exists i \in \{1, \dots, N_r(2m, lm)\} \mid \text{Diam}(\mathcal{G}_{2m}(i, lm)) \le c' f(2m)\} \right) \le \frac{1}{4},
+$$
+
+so that
+
+$$
+\mathbb{P}(\mathrm{Diam}(\mathcal{G}_r) \leq m/3 \wedge c \cdot c'(r/m)^2 f(2m)) \leq \frac{1}{4},
+$$
+
+which implies the desired bound, by the definition of *f*.
+
+The details of the proof can be adapted from the proof of [14, Proposition 1]. $\square$
+
+Theorem 7.2 is then a purely analytic consequence of Proposition 7.4, and its proof is
+a straightforward adaptation of that of [14, Theorem 5].
+
+We can now use Theorem 7.2 to obtain the following lower bounds for the distances
+along the boundary of $\mathcal{L}$:
+
+**Proposition 7.5.** For every $\varepsilon > 0$, there exists an integer $K > 0$ such that, for every $r \ge 1$,
+
+$$
+\mathbb{P} \left( \min_{|j| \ge K r^2} \vec{d}_{\mathcal{L}}((0,0), (j,0)) \ge r \right) \ge 1 - \varepsilon.
+$$
+
+Consequently, for $K' = 9K$, we also have, for every $r \ge 1$,
+
+$$
+\mathbb{P} \left( \min_{|j| \ge 2K'r^2} \min_{-K'r^2 \le i \le K'r^2} \vec{d}_{\mathcal{L}}((i,0), (j,0)) \ge r \right) \ge 1 - 2\varepsilon.
+$$
+
+Proof. Let us start with the first assertion. Let $\epsilon > 0$. Fix $r \ge 1$, and $K \ge 1$. Then, from (5.18), the number $N_{(K,r)}$ of trees that reach height $r$ between $(0,0)$ and $(j,0)$ is bounded below by a binomial variable of parameters $(Kr^2, 3/((r+2)^2-1))$, so that, using Chebyshev's inequality, for any $a > 0$,
+
+$$
+\mathbb{P}\left(N_{(K,r)} \le \frac{3}{8}K - a\right) \le \frac{3K}{a^2}.
+$$
+
+(Note that the binomial variable in question has expectation greater than or equal to
+$3K/8$, with equality when $r=1$, and a variance smaller than $3K$.)
+
+Taking $a = \sqrt{(6K/\epsilon)}$, for $K$ large enough that $a \le (1/8)K + 1$, we get
+
+$$
+\mathbb{P}\left(N_{(K,r)} \le \frac{1}{4}K + 1\right) \le \frac{\epsilon}{2}. \tag{7.1}
+$$
+
+Now, on the event that $N_{(K,r)} > K/4$, for any $j \ge Kr^2$, we have
+
+$$
+\vec{d}_{\mathcal{L}}((0,0), (j,0)) \geq \sum_{i=1}^{\lfloor \frac{K}{4} \rfloor + 1} \mathrm{Diam}(\mathcal{G}_r(i)) \wedge r,
+$$
+
+so that, using Theorem 7.2,
+
+$$
+\mathbb{P}\left(\vec{d}_{\mathcal{L}}((0,0), (j,0)) < c r \frac{K}{4} \wedge r\right) \leq \frac{1}{2^{K/4}}.
+$$
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diff --git a/samples/texts/2952501/page_35.md b/samples/texts/2952501/page_35.md
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+The details of the proof can be adapted verbatim from the proof of Proposition 17 in [13]. □
+
+# 8 Asymptotic equivalence between oriented and non-oriented distances
+
+Recall that, on any Eulerian triangulation with a boundary $A$, we write $\vec{d}_A$ for the oriented distance on $A$, and $d_A$ for the usual graph distance. We will show that these two distances are asymptotically proportional, first on the layers of the LHPET $\mathcal{L}$, then on the ones of the UIPET $\mathcal{T}_{\infty}^{(1)}$, and finally in large finite Eulerian triangulations. To do so, we follow the chain of proofs of Sections 5 and 6 in [13], once again detailing mostly the additional arguments needed in our case.
+
+## 8.1 Subadditivity in the LHPET and the UIPET
+
+Recall that we write $\rho$ for the root vertex (0, 0) of the LPHET $\mathcal{L}$, and that $\mathcal{L}_r$ is the lower boundary of the layer $\mathcal{L}_{[0,r]}$. We have the following result:
+
+**Proposition 8.1.** There exists a constant $c_0 \in [2/3, 1]$ such that
+
+$$r^{-1} d_{\mathcal{L}}(\rho, \mathcal{L}_r) \xrightarrow[r \to \infty]{a.s.} c_0.$$
+
+*Proof.* The proof of this result, apart from the bounds on $c_0$, is essentially the same as that of [13, Proposition 18]. However, as it is a very short argument, but central in this whole work, we write it here in its entirety.
+
+For integers $0 \le m < n$, recall that $\mathcal{L}_{[m,n]}$ is the infinite planar map obtained by keeping only the layers of $\mathcal{L}$ between the levels $m$ and $n$. The non-oriented distance $d_{\mathcal{L}_{[m,n]}}$ on this strip is defined by considering the shortest non-oriented paths that stay in $\mathcal{L}_{[m,n]}$. Thus, for two vertices $v, v' \in \mathcal{L}_{[m,n]}$, we have $d_{\mathcal{L}_{[m,n]}}(v, v') \ge d_{\mathcal{L}}(v, v')$.
+
+Let then $m, n \ge 1$, and let $x_m$ be the leftmost vertex $x$ of $\mathcal{L}_m$ such that $d_{\mathcal{L}}(\rho, \mathcal{L}_m) = d_{\mathcal{L}}(\rho, x)$. We have
+
+$$d_{\mathcal{L}}(\rho, \mathcal{L}_{m+n}) \le d_{\mathcal{L}}(\rho, \mathcal{L}_m) + d_{\mathcal{L}_{[m,m+n]}}(x_m, \mathcal{L}_{m+n}).$$
+
+As $x_m$ is a function of $\mathcal{L}_{[0,m]}$ only, and the layers in $\mathcal{L}$ are independent, the random variable $d_{\mathcal{L}_{[m,m+n]}}(x_m, \mathcal{L}_{m+n})$ is independent of $\mathcal{L}_{[0,m]}$, and has the same distribution as $d_{\mathcal{L}}(\rho, \mathcal{L}_n)$.
+
+We can then apply Liggett's version of Kingman's subadditive theorem [24], to get the desired convergence: the fact that the limit is a constant follows from Kolmogorov's zero-one law. As for the bounds for $c_0$, it is clear from (2.1) that $c_0 \in [1/2, 1]$. Our proof that $c_0$ must be at least 2/3 relies on a result of asymptotic proportionality in finite Eulerian triangulations, that will be stated further in Theorem 1.2. We thus postpone this argument to after Theorem 1.2. □
+
+To carry this asymptotic proportionality over to large finite Eulerian triangulations, we will make a stop at the UIPET of the digon $\mathcal{T}_{\infty}^{(1)}$. In the remainder of this subsection, we write $d$ for the non-oriented distance on $\mathcal{T}_{\infty}^{(1)}$, $B_n^{\bullet}$ for $B_n^{\bullet}(\mathcal{T}_{\infty}^{(1)})$ and $\partial^* B_n^{\bullet}$ for $\partial^* B_n^{\bullet}(\mathcal{T}_{\infty}^{(1)})$, to simplify notation.
+
+**Proposition 8.2.** Let $\epsilon, \delta \in (0, 1)$. We can find $\eta \in (0, 1/2)$ such that, for every sufficiently large $n$, the property
+
+$$ (1 - \epsilon)\mathbf{c}_0\eta n \le d(v, \partial^* B_{n-\lfloor\eta n\rfloor}^{\bullet}) \le (1 + \epsilon)\mathbf{c}_0\eta n \quad \forall v \in \partial^* B_n^{\bullet} $$
+
+holds with probability at least $1 - \delta$.
\ No newline at end of file
diff --git a/samples/texts/2952501/page_37.md b/samples/texts/2952501/page_37.md
new file mode 100644
index 0000000000000000000000000000000000000000..0481b752dd92254272123a7706e0debdaf402513
--- /dev/null
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@@ -0,0 +1,35 @@
+a distinguished vertex $o_n$. The hull $B_r^\bullet(\bar{T}_n^{(1)})$ is well-defined when $\vec{d}(\rho_n, o_n) > r + 1$, otherwise we set it to be $\bar{T}_n^{(1)}$.
+
+**Lemma 8.5.** There exists a constant $\bar{c} > 0$ such that, for every $n,r,p \ge 1$ and every $\Delta \in C_{1,r}$ with top boundary half-length $p$, such that $n > N(\Delta) + p$,
+
+$$ P(B_r^\bullet(\bar{T}_n^{(1)}) = \Delta) \le \bar{c} \left( \frac{n}{n - N(\Delta) + 1} \right)^{3/2} \cdot P(B_r^\bullet(\mathcal{T}_\infty^{(1)}) = \Delta). \quad (8.1) $$
+
+*Proof.* The proof of this lemma is very similar to that of [13, Lemma 22], with the additional subtlety that, like for Lemma 5.3, we do not start with explicit expressions for probabilities in finite triangulations, as shown in (8.3).
+
+Fix $r \ge 1$ and $\Delta \in C_{1,r}$ with top boundary half-length $p$. We will write $V$ for $\#V(\Delta)$ to simplify notation. Using (5.4) and the fact that $\mathcal{T}_\infty^{(1)}$ is the local limit of $\mathcal{T}_n^{(1)}$, we have
+
+$$ P(B_r^{\bullet}(\mathcal{T}_{\infty}^{(1)}) = \Delta) = \frac{C(p)}{C(1)} 8^{-N(\Delta)}. \qquad (8.2) $$
+
+On the other hand, (5.3) gives the formula
+
+$$ P \left( B_r^\bullet (\bar{T}_n^{(1)}) = \Delta \right) = \frac{B_{n-N,p}}{B_{n,1}} \cdot \frac{\#\text{inner vertices in } \mathcal{T}_n^{(1)} \setminus \Delta}{\#\text{inner vertices in } \mathcal{T}_n^{(1)}} \quad (8.3) $$
+
+$$ \leq \frac{B_{n-N,p}}{B_{n,1}} \cdot \frac{n-V}{n}, \quad (8.4) $$
+
+where the last inequality is given by Euler's formula and the fact that at most $p$ vertices of $\partial^*\Delta$ are identified together in $\mathcal{T}_n^{(1)}$. (We still need $n > N + p$ since $\mathcal{T}_n^{(1)} \setminus \Delta$ will have $n - N - p$ inner vertices if none of these identifications occur.)
+
+Then, using the bounds of (4.5) and the asymptotics of (4.3), we get that
+
+$$ P(B_r^{\bullet}(\overline{\mathcal{T}}_n^{(1)}) = \Delta) \le c^* C(p) \left(\frac{n}{n-N}\right)^{3/2} 8^{-N} $$
+
+for some constant $c^*$. Comparing the last bound with (8.2) gives the desired result. $\square$
+
+*Proof of Proposition 8.4.* Fix $\varepsilon > 0$ and $\nu > 0$. Its suffices to prove that, for all $n$ sufficiently large, we have
+
+$$ P\left(\left|\frac{d(\rho_n, o_n)}{\vec{d}(\rho_n, o_n)} - c_0\right| > 2\varepsilon\right) < \nu. \quad (8.5) $$
+
+Indeed, as detailed in Proposition 4.1, the sequence $n^{-1/4}\vec{d}(\rho_n, o_n)$ is bounded in probability, so that the statement of the proposition will follow from (8.5). Note that, in [13], the equivalent tightness is obtained as a consequence of the convergence of usual planar triangulations to the Brownian map: in our case, we had to use the weaker result of Theorem 2.12, as we obviously do not have a convergence at the level of maps yet.
+
+To obtain (8.5), we want to transfer the results of Proposition 8.3 on the UIPET to large finite triangulations. This necessitates the bounds of Lemma 8.5, together with the statement of Proposition 4.2 on the profile of distances in large finite triangulations (which is once again a consequence of Theorem 2.12, as we cannot rely on a convergence to the Brownian map).
+
+We omit the details of the proof of (8.5), as they can be straightforwardly adapted from the equivalent statement in the proof of Proposition 21 in [13], replacing once again $d_{gr}$ by $\vec{d}$, and $d_{fpp}$ by $d$. $\square$
\ No newline at end of file
diff --git a/samples/texts/2952501/page_38.md b/samples/texts/2952501/page_38.md
new file mode 100644
index 0000000000000000000000000000000000000000..6421b3d97415c2a88e94af74ad169f5bc857e068
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+We will now derive our final result of asymptotic proportionality between the oriented and non-oriented distances, Theorem 1.2. This one is in the context of $\mathcal{T}_n$, the uniform rooted plane Eulerian triangulation with $n$ black faces, that is in correspondence with $\mathcal{T}_n^{(1)}$ as shown in Figure 10. As previously, we use $d$ to denote the non-oriented graph distance.
+
+*Proof of Theorem 1.2.* Let us give an idea of the proof of this theorem, which follows the arguments of the proof of Theorem 1 in [13].
+
+From Proposition 8.4 and the correspondence between $\mathcal{T}_n^{(1)}$ and $\mathcal{T}_n$, we get that, if $o_n'$ is a uniform vertex of $\mathcal{T}_n$, we have
+
+$$ \mathbb{P}(|d(\rho_n, o_n') - c_0 \vec{d}(\rho_n, o_n')| > \varepsilon n^{1/4}) \xrightarrow{n \to \infty} 0. \qquad (8.6) $$
+
+Observe now that $\bar{\mathcal{T}}_n$, re-rooted at $\rho_n'$, the origin vertex of a random uniform edge $e_n$ (remember that all edges of $\mathcal{T}_n$ have a canonical orientation), still pointed at $o_n'$, has the same distribution as $\bar{\mathcal{T}}_n$. This implies that the statement of (8.6) also holds for the distances from $\rho_n'$, that is sampled according to its degree:
+
+$$ \mathbb{P}(|d(\rho'_n, o'_n) - c_0 \vec{d}(\rho'_n, o'_n)| > \varepsilon n^{1/4}) \xrightarrow{n \to \infty} 0. $$
+
+As the numbers of edges and vertices of $\mathcal{T}_n$ are fixed, this allows us to deduce a similar statement on distances between two random uniform vertices $o_n'$, $o_n''$ of $\mathcal{T}_n$:
+
+$$ \mathbb{P}\left(|d(o_n', o_n'') - c_0 \vec{d}(o_n', o_n'')| > \varepsilon n^{1/4}\right) \xrightarrow{n \to \infty} 0. \qquad (8.7) $$
+
+We now want to make this statement into a global one on all the vertices of $\mathcal{T}_n$.
+
+Let us fix $\delta \in (0, 1/2)$. We can choose an integer $k \ge 1$ such that, for every $n$ sufficiently large, we can pick $k$ random vertices $(o_n^1, \dots, o_n^k)$ uniformly in $\mathcal{T}_n$ and independently from one another, satisfying
+
+$$ \mathbb{P}\left(\sup_{x \in V(\mathcal{T}_n)} \left(\inf_{1 \le j \le k} \vec{d}(x, o_n^j)\right) < \varepsilon n^{1/4}\right) > 1 - \delta. \qquad (8.8) $$
+
+This follows from Proposition 4.3. Note that, once again, the equivalent property in [13] was obtained as a consequence of the convergence of usual planar triangulations to the Brownian map, whereas here we had to obtain it from the convergence of the rescaled oriented distances from $o_n$ to a Brownian snake, which is a weaker result.
+
+Then, (8.7) implies that we also have, for all sufficiently large $n$,
+
+$$ \mathbb{P}\left(\bigcap_{1 \le i \le j \le k} \{|d(o_n^i, o_n^j) - c_0 \vec{d}(o_n^i, o_n^j)| \le \varepsilon n^{1/4}\}\right) > 1 - \delta. $$
+
+Observe now that
+
+$$ \sup_{x,y \in V(\mathcal{T}_n)} |d(x,y) - c_0 \vec{d}(x,y)| \le \sup_{1 \le i,j \le N} |d(o_n^i, o_n^j) - c_0 \vec{d}(o_n^i, o_n^j)| + 5 \sup_{x \in V(\mathcal{T}_n)} \left( \inf_{1 \le j \le N} \vec{d}(x,o_n^j) \right). $$
+
+Using the previous two bounds, the right-hand side of this inequality can be bounded by $6\varepsilon$ outside a set of probability at least $2\delta$ for all sufficiently large $n$, which concludes the proof. □
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+Let us finally give a short proof of why $c_0 \ge 2/3$. Consider $\mathcal{T}_n$, the uniform rooted plane Eulerian triangulation with $n$ black faces. From Theorem 1.2, for any $\varepsilon, \delta \in (0, 1/2)$, for any $n$ large enough,
+
+$$|d_n(x, y) - c_0 \vec{d}_n(x, y)| \le \varepsilon n^{1/4}, \quad \forall x, y \in V(\mathcal{T}_n), \qquad (8.9)$$
+
+outside an event of probability less than $\delta$.
+
+Suppose that $c_0 < 2/3$. Let us fix $n \ge 1$, and consider some $c \in (0, 1)$. Then, on the event of (8.9), for any $x, y \in V(\mathcal{T}_n)$ such that
+
+$$d_n(x, y) \ge cn^{1/4}, \qquad (8.10)$$
+
+we have:
+
+$$\vec{d}_n(x, y) \ge \left( \frac{1}{c_0} - \frac{\varepsilon}{c} \right) d_n(x, y).$$
+
+This means that, for any geodesic $\gamma$ for the distance $d_n$ from $x$ to $y$ in $\mathcal{T}_n$, a fraction larger than or equal to $(1/c_0 - 1 - \varepsilon/c)$ of the edges of $\gamma$ are oriented from $y$ to $x$. But, as the above bound also applies when we exchange $x$ and $y$, a same fraction of edges of $\gamma$ must be oriented from $x$ to $y$, which is not possible if $(1/c_0 - 1 - \varepsilon/c) > 1/2$, that is, $\varepsilon/c < 1/c_0 - 3/2$.
+
+Since, for any $\delta \in (0, 1/2)$, there exists a $c(\delta) \in (0, 1)$ such that, if $n$ is large enough, outside of an event of probability less than $\delta$, a positive proportion of pairs of vertices of $\mathcal{T}_n$ satisfy (8.10), we deduce that (8.9) cannot have a high probability for large $n$, if $c_0 < 2/3$.
+
+It would be interesting to also refine the upper bound on $c_0$. However, this seems to necessitate deeper arguments than our refinement of the lower bound.
+
+# 9 Convergence for the Riemannian distance
+
+We now turn our attention to another distance that can be defined on Eulerian triangulations, the **Riemannian distance** $d^R$. To define this distance, we start by assigning to a triangulation $A$, the piecewise-linear metric space $S(A)$, obtained by gluing equilateral, Euclidean triangles with sides of unit length, according to the combinatorics of $A$. We call this space the **Euclidean geometric realization** of $A$. It naturally comes endowed with a metric, that we denote by $d^R$, and, by a slight abuse of notation, we also denote by $d^R$ the induced distance on the vertices of $A$.
+
+We want to show that, like the usual graph distance $d$, the Riemannian distance $d^R$ is asymptotically proportional to the oriented distance $\vec{d}$, so that, endowed with $d^R$, the uniform Eulerian triangulation $\mathcal{T}_n$ still converges to the Brownian map. This can be once again proven using the layer decomposition of finite and infinite Eulerian triangulations with respect to $\vec{d}$, together with an ergodic subadditivity argument. Once this argument gives the desired asymptotic proportionality on $\mathcal{L}$, the results of Section 8 can be directly adapted to $d^R$, to obtain the new convergence to the Brownian map.
+
+However, the subbadditivity argument presents here a hurdle that was not present in the case of $d$: indeed, while we still have the immediate upper bound $d^R \le \vec{d}$ (and even: $d^R \le d$), we have no obvious way to bound $d^R$ from below with $\vec{d}$. Such a bound is crucial, since, without it, the proportionality constant given by the ergodic subbadditivity theorem could very well be zero. We therefore prove the following result:
+
+**Proposition 9.1.** Let $A$ be a triangulation, endowed with its graph distance $d$, canonical oriented pseudo-distance $\vec{d}$ and Riemannian distance $d^R$. Then,
+
+$$d^R \ge \frac{\sqrt{3}}{4} d.$$
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diff --git a/samples/texts/2952501/page_4.md b/samples/texts/2952501/page_4.md
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+Figure 6: Two disjoint curves in $C_n(A)$ cannot encircle one other.
+
+**Lemma 2.13.** Let A be a planar rooted Eulerian triangulation. For a given vertex v at (oriented) distance at least $n + 1$ from the root, there is a unique curve in $C_n(A)$ that separates v from the root.
+
+Moreover, all curves in $C_n(A)$ are simple.
+
+*Proof.* First consider two disjoint curves in $C_n(A)$ that separate the same vertex v from the origin. Necessarily, a geodesic path from the root to a vertex belonging to one of them should go through the other, and thus have length at least $n + 1$ (see Figure 6).
+
+Now, if two curves of $C_n(A)$ intersect at a vertex of type $n$, then by our resolution rule, they cannot go counterclockwise around the same region of A (see Figure 7), and, as explained before, these are precisely the regions they separate from the origin.
+
+This rule also implies that a curve $\mathcal{C}$ in $C_n(A)$ cannot go twice through the same type-$n$ vertex. Indeed, if that were the case, then $\mathcal{C}$ would separate from the origin vertices of oriented distance $n-1$ and less, so that any oriented geodesic from the origin to these vertices should be of length at least $n+1$ (see Figure 7). $\square$
+
+We define the **ball** $B_n(A)$ as the submap of A obtained by keeping only the faces and edges of A incident to at least a vertex at distance $n-1$ or less from the origin, cutting along the edges of type $n \to n+1$, and filling in the produced holes by simple faces (see Figure 8 for a local depiction of this procedure). Thus, in $B_n(A)$, for each closed curve $\mathcal{C} \in C_n(A)$, we have replaced all faces that $\mathcal{C}$ separates from the root, by a single, simple face. In particular, if two faces of A of type $n$ share a type-( $n \to n+1$ ) edge, in $B_n(A)$ their respective type-( $n \to n+1$ ) edges are not identified, so that their common type-( $n+1$ ) vertex gives rise to two vertices in $B_n(A)$ (see Figure 9). Two type-$n$ faces $f, f'$ may also share a type-( $n+1$ ) vertex $v$ but no edge: in that case, it means that $v$ is also shared by faces of types $n+1$, so that we would need to add these faces and the type $n \to n+1$ edges they share with $f$ and/or $f'$, in order to identify the type-( $n+1$ ) vertices of $f$ and $f'$ into $v$. Note that as $B_n(A)$ contains all the type-( $n-1 \to n$ ) edges of A, type-$n$ vertices of A are never duplicated in $B_n(A)$.
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diff --git a/samples/texts/2952501/page_41.md b/samples/texts/2952501/page_41.md
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+**Proposition 9.2.** There exists a constant $\mathbf{c}_1 \in [(\sqrt{3}/4)\mathbf{c}_0, 1]$ such that
+
+$$r^{-1} d_{\mathcal{L}}^{R}(\rho, \mathcal{L}_r) \xrightarrow[r \to \infty]{a.s.} \mathbf{c}_1.$$
+
+In the sequel, $\mathbf{c}_1$ will refer to the constant of Proposition 9.2.
+
+*Proof.* We proceed like for Proposition 8.1, by considering the layers $\mathcal{L}_{[m,n]}$, for integers $0 \le m \le n$. Such a layer corresponds to a strip in the Euclidean geometrical realization $S(\mathcal{L})$ of $\mathcal{L}$: we can define the Riemannian distance $d_{\mathcal{L}_{[m,n]}}^R$ on the vertices of $\mathcal{L}_{[m,n]}$ by considering the shortest paths (**starting and ending at vertices**) in $S(\mathcal{L})$ that stay in this strip. Then, we have, for any two vertices $v, v' \in \mathcal{L}[m, n]$, we have $d_{\mathcal{L}_{[m,n]}}^R(v, v') \ge d_{\mathcal{L}}^R(v, v')$.
+
+Thus, as for the graph distance, if $m, n \ge 1$, and $x_m$ is the leftmost vertex $x$ of $\mathcal{L}_m$ such that $d_{\mathcal{L}}^R(\rho, \mathcal{L}_m) = d_{\mathcal{L}}^R(\rho, x)$, we have
+
+$$d_{\mathcal{L}}^{R}(\rho, \mathcal{L}_{m+n}) \leq d_{\mathcal{L}}^{R}(\rho, \mathcal{L}_{m}) + d_{\mathcal{L}_{[m,n]}}^{R}(x_m, \mathcal{L}_{m+n}).$$
+
+As in the case of the graph distance $d$, since $x_m$ is a function of $\mathcal{L}_{[0,m]}$ only, and since the layers in $\mathcal{L}$ are i.i.d., this yields the desired convergence.
+
+The upper bound on $\mathbf{c}_1$ is immediate; let us briefly explain how we obtain the lower bound. Fix $\varepsilon > 0$ and $\delta \in (0, 1/2)$. We have that, for $n$ large enough, outside of an event of probability less than $\delta$,
+
+$$|d_n(x, y) - \mathbf{c}_0 \vec{d}_n(x, y)| \leq \varepsilon n^{1/4} \quad \forall x, y \in V(\mathcal{T}_n),$$
+
+so that we get from Proposition 9.1:
+
+$$d_n^R(x, y) \geq \frac{\sqrt{3}}{4} \mathbf{c}_0 \vec{d}_n(x, y) - \varepsilon n^{1/4}.$$
+
+Now, there exists a constant $0 < C(\delta) < 1$, that depends only on $\delta$, such that, for $n$ large enough, outside of an event of probability less than $\delta$, a positive proportion of pairs of vertices of $\mathcal{T}_n$ satisfy
+
+$$\vec{d}_n(x, y) \geq C(\delta)n^{1/4}.$$
+
+Therefore, for all such pairs, we have
+
+$$d_n^R(x, y) \geq \left( \frac{\sqrt{3}}{4} \mathbf{c}_0 - \frac{\varepsilon}{C(\delta)} \right) \vec{d}_n(x, y),$$
+
+so that, necessarily,
+
+$$\mathbf{c}_1 \geq \frac{\sqrt{3}}{4} \mathbf{c}_0. \quad \square$$
+
+Retracing for $d^R$ the same arguments as the ones used for $d$ in Section 8, we deduce from Proposition 9.2 the following result:
+
+**Theorem 9.3.** Let $\mathcal{T}_n$ be a uniform random rooted Eulerian planar triangulation with $n$ black faces, and let $V(\mathcal{T}_n)$ be its vertex set. For every $\varepsilon > 0$, we have
+
+$$\mathrm{P}\left(\sup_{x,y \in V(\mathcal{T}_n)} |d_n^R(x,y) - c_1 \vec{d}_n(x,y)| > \varepsilon n^{1/4}\right) \xrightarrow[n \to \infty]{} 0.$$
+
+This allows us to add a third scaling limit to the joint convergence of Theorem 3.1:
+
+**Corollary 9.4.** Let $(\mathbf{m}_\infty, D^*)$ be the Brownian map. We have the following joint convergences
+
+$$\begin{align*}
+& n^{-1/4} \cdot (V(\mathcal{T}_n), \overleftrightarrow{d}_n) \xrightarrow[n \to \infty]{(d)} (\mathbf{m}_\infty, D^*) \\
+& n^{-1/4} \cdot (V(\mathcal{T}_n), d_n) \xrightarrow[n \to \infty]{(d)} \mathbf{c}_0 \cdot (\mathbf{m}_\infty, D^*) \\
+& n^{-1/4} \cdot (S(\mathcal{T}_n), d_n^R) \xrightarrow[n \to \infty]{(d)} \mathbf{c}_1 \cdot (\mathbf{m}_\infty, D^*),
+\end{align*}$$
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+for the Gromov-Hausdorff distance on the space of isometry classes of compact metric spaces.
+
+Let us sketch very quickly the proof of Corollary 9.4: following the same steps as the ones we made for $d$ in Section 8, the result of Theorem 9.3 implies that the Brownian map is the scaling limit of the vertex set $V(T_n)$ endowed with the distance induced by $S(T_n)$, and not $S(T_n)$ itself. However, the Gromov-Hausdorff distance between $(S(T_n), d^R)$ and $(V(T_n), d^R)$ is at most $\sqrt{3}/4$ (considering $V(T_n)$ as embedded into $S(T_n)$), so that this convergence does extend to $S(T_n)$.
+
+As explained in the introduction, the result of Corollary 9.4 allows us to make a more direct comparison between models of random maps as studied by probabilists, and models of 2D quantum gravity studied by theoretical physicists, such as Causal Dynamical Triangulations, as the latter models focus on the Euclidean geometric realization associated to some combinatorial maps.
+
+Note that the geometric argument in the proof of Proposition 9.1 works for any triangulation, and not just an Eulerian one. Thus, relying on the layer decomposition of usual triangulations of [13], we can prove in the same way as here that usual triangulations, equipped with the Riemannian metric, also converge to the Brownian map. A similar geometric argument should also work for quadrangulations, along with the layer decomposition of [21].
+
+## References
+
+[1] C. Abraham, *Rescaled bipartite planar maps converge to the Brownian map*, Ann. Inst. Henri Poincaré Probab. Stat. **52** (2016), no. 2, 575–595. MR-3498001
+
+[2] L. Addario-Berry and M. Albenque, *Convergence of odd-angulations via symmetrization of labeled trees*, arXiv:1904.04786
+
+[3] L. Addario-Berry and M. Albenque, *The scaling limit of random simple triangulations and random simple quadrangulations*, Ann. Probab. **45** (2017), no. 5, 2767–2825. MR-3706731
+
+[4] M. Albenque and J. Bouttier, *Constellations and multicontinued fractions: application to Eulerian triangulations*, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), Discrete Math. Theor. Comput. Sci. Proc., AR, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012, pp. 805–816. MR-2958050
+
+[5] J. Ambjørn, A. Görlich, J. Jurkiewicz, and R. Loll, *Quantum gravity via causal dynamical triangulations*, Springer handbook of spacetime, Springer, Dordrecht, 2014, pp. 723–741. MR-3525043
+
+[6] O. Angel, *Growth and percolation on the uniform infinite planar triangulation*, Geom. Funct. Anal. **13** (2003), 935–974. MR-2024412
+
+[7] J. Bettinelli, E. Jacob, and G. Miermont, *The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection*, Electron. J. Probab. **19** (2014), no. 74, 16. MR-3256874
+
+[8] M. Bousquet-Mélou and G. Schaeffer, *Enumeration of planar constellations*, Adv. in Appl. Math. **24** (2000), no. 4, 337–368. MR-1761777
+
+[9] J. Bouttier and A. Carrance, *Enumeration of hypermaps with alternating boundaries*, arXiv:2012.14258
+
+[10] J. Bouttier, P. Di Francesco, and E. Guitter, *Planar maps as labeled mobiles*, Electr. J. Combin. **11** (2004), no. 1. MR-2097335
+
+[11] D. Burago, Y. Burago, and S. Ivanov, *A course in metric geometry*, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, 2001. MR-1835418
+
+[12] A. Carrance, *Triangulations colorées aléatoires*, Ph.D. thesis, Université Claude Bernard Lyon 1, 2019.
+
+[13] N. Curien and J.-F. Le Gall, *First-passage percolation and local modifications of distances in random triangulations*, Ann. Sci. ENS **52** (2019), 631–701. MR-3982872
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diff --git a/samples/texts/2952501/page_44.md b/samples/texts/2952501/page_44.md
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+For any map or graph $G$, we will denote by $V(G)$ its vertex set.
+In this paper, we will come upon two specific types of maps:
+
+**Definition 2.5.** A **tree** is a connected graph with no cycle. A **plane tree** is a map $T$ that, as a graph, is a tree. Since $T$ has no cycle, it is necessarily a planar map.
+
+**Definition 2.6.** An **Eulerian triangulation** is a map whose faces have all degree 3, and such that these faces can be properly bicolored, i.e., colored in black and white, such that all white faces are only adjacent to black faces, and vice versa.
+
+We will also deal with **Eulerian triangulations with a boundary**, that is, maps with one distinguished face, such that all its other faces have degree 3, and these inner faces can be properly bicolored (i.e., colored in black and white, such that white faces are only adjacent to black faces or to the external face, and similarly for black faces).
+
+By convention, when we root an Eulerian triangulation with a boundary, we do so on an edge adjacent to the external face.
+
+## 2.2 Bijection with trees
+
+We consider here rooted, planar Eulerian triangulations. Bouttier, Di Francesco and Guitter [10] have established a bijection between this family of maps and a particular class of labeled trees, whose construction we now briefly recall and extend.
+
+Let $A$ be a rooted planar Eulerian triangulation. The orientation of the root edge of $A$ fixes a **canonical orientation** of all its edges, by requiring that orientations alternate around each vertex. By construction, edges around a given face are necessarily oriented either all clockwise, or all anti-clockwise (with respect to the plane embedding in which the face on the left of the root edge is the infinite one). This fixes the bicoloration of the faces of $A$, by setting for instance that clockwise faces are black, and anti-clockwise faces, white.
+
+From now on, any mention of orientation refers to this canonical orientation.
+
+Figure 1: A planar Eulerian triangulation with its canonical orientation, bicoloration and oriented geodesic distances.
+
+For any pair $(u, v)$ of vertices of $A$, we define the **oriented distance** $\vec{d}(u, v)$ from $u$ to $v$, as the minimal length of an oriented path from $u$ to $v$.
+
+Let us state a useful fact. Denoting by $d$ the usual graph distance, in any Eulerian triangulation, we always have:
+
+$$d \leq \vec{d} \leq 2d, \tag{2.1}$$
+
+as, in the worst case, the oriented distance forces a path to go through two edges of a triangle instead of just taking the third one.
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+face into a number of white faces. By construction, the clockwise sequence of labels
+around any of these white faces is of the form $0 \rightarrow 2 \rightarrow \dots \rightarrow 2 \rightarrow 1$, where all the labels
+between the first and last "2" are greater or equal to 2, and all increments but the first
+are $\pm 1$. For each white face $F$ that is not already a triangle, and for each type-$ (2 \rightarrow 3)$
+edge whose right side is adjacent to $F$, we create a black triangle of type 2 to the right of
+this edge, by adding edges between its vertices and the unique vertex labeled 1 around
+$F$. This induces a splitting of $F$ into smaller white faces, and we repeat the procedure
+again, until all labels are exhausted. This yields an Eulerian triangulation $A$ rooted at
+the $0 \rightarrow 1$ edge linking the origin to the root of $T$ (see Figure 3).
+
+Figure 3: The inverse construction of the triangulation of Figure 1 from the labeled tree.
+
+In the sequel, it will be more convenient to deal with trees whose labels are not
+necessarily positive. For that purpose, we point $A$ at some vertex $v_*$, and consider the
+new labeling $\ell$ on $V(A)$ given by:
+
+$$\ell(u) = \vec{d}(v_*, u) - \vec{d}(v_*, \rho).$$
+
+Let us proceed with this new labeling as we did with the oriented geodesic distance: starting from the triangulation $A$, in each black triangle of type $n+1$ (where now the integer $n$ may be nonpositive), we only keep the edge of type $n+1 \rightarrow n+2$. This
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diff --git a/samples/texts/2952501/page_47.md b/samples/texts/2952501/page_47.md
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+construction now gives a correspondance between pointed planar Eulerian triangulations with $n$ black triangles, and well-labeled trees with $n$ edges, with no constraint on the label signs, that still maps the vertices of the triangulation, minus the distinguished vertex, to those of the tree. Let us now pay attention to the rooting: since the distinguished vertex that we add to the tree is no longer the origin of the triangulation (and, conversely, the root corner of the tree is no longer at a minimal label), we need additional information with either object, to know how to root the other. Thus, in the triangulation-to-tree direction, we start from a couple $(A, \varepsilon)$, with $\varepsilon \in \{0, 1\}$: depending on the value of $\varepsilon$, we root the tree $T$ at the edge remaining from the (black) root face of $A$, either with its original direction, or the reverse one, see Figure 4. Note that we have to shift the labels of $T$ by an integer $L(A)$ between -2 and 2, so that the root corner has label 1, so that the labeling of the vertices of the tree is:
+
+$$l(u) = \vec{d}(v_*, u) - \vec{d}(v_*, \rho) + L(A).$$
+
+Conversely, in the tree-to-triangulation direction, we start from a couple $(T, \delta)$, with $\delta \in \{0, 1, 2\}$: depending on the value of $\delta$, the root edge of $A$ is either the type-$n-1 \to n$, $n \to n+1$, or $n+1 \to n-1$ edge of the black face adjacent to the root edge of $T$ (where this face is itself of type $n$).
+
+It is straightforward to prove similarly to the case of $\varphi_n$, that this new mapping is a bijection as well:
+
+**Proposition 2.10.** Let us denote by $\bar{\mathcal{T}}_n$ the set of pointed, rooted planar Eulerian triangulations with $n$ faces, and by $\mathbb{T}_n$ the set of well-labeled trees with $n$ edges. The mapping $\psi_n$ detailed above from $\bar{\mathcal{T}}_n \times \{0, 1\}$ to $\mathbb{T}_n \times \{0, 1, 2\}$ is a bijection.
+
+Consider now the random variable $(\mathcal{T}_n^\bullet, \varepsilon_n)$, where $\mathcal{T}_n^\bullet$ is picked uniformly at random in $\bar{\mathcal{T}}_n$, and $\varepsilon_n$ is uniform over $\{0, 1\}$. Since these additional decorations of $\psi_n$ only influence the rooting of the obtained map, the image of $(\mathcal{T}_n^\bullet, \varepsilon_n)$ can be written as $(\mathcal{T}_n, \delta_n)$, where $\mathcal{T}_n$ is uniform over $\mathbb{T}_n$, and $\delta_n$ is uniform over $\{0, 1, 2\}$.
+
+Furthermore, note that, if $T = \psi_n(A)$, for every vertex $u$ of $T$, the oriented distance from $v_*$ to $u$ in $A$ is given by:
+
+$$\vec{d}(v_*, u) = l(u) - \min_{v \in V(T)} l(v) + 1. \quad (2.2)$$
+
+To get more general information on the oriented distances in $A$ from the labels of $T$, we need a bit of additional notation.
+
+First observe that, with the construction of $A$ from $T$, a corner $c$ of $T$ is always incident in $A$ to an edge oriented from the first corner $c'$ encountered when going anticlockwise around the unique face of $T$, starting at $c$, and that has label $l(c) - 1$ (by convention, we define the label of a corner in $T$ to be the label of the associated vertex). Indeed, either this corner was already adjacent to $c$ in $T$, or we create an edge between them when adding a black triangle to the right of the edge of type $l(c) \to l(c) + 1$ that starts at $c$.
+
+We call $c'$, the **predecessor** of $c$, and denote it by $p(c)$. (The predecessor of a corner of minimal label is naturally the corner of the origin to which it is linked in the first step of the construction.) We also call $p^k(c)$ the $k$-th predecessor of $c$, whenever it is defined.
+
+Let us denote by $l_0$ the minimal label of the vertices of $T$. For a corner $c$ of $T$, the edge $p(c) \to c$ in $A$ is obviously of type $l(c) - l_0 \to l(c) - l_0 + 1$ (for the oriented distance from $v_*$). This implies that the path from $v_*$ to $c$ going through all its predecessors: $v_* \to p(l(c)-1)(c) \to \dots \to p(c) \to c$ is a geodesic for $\vec{d}$ in $A$.
+
+For any pair of corners $c, c'$ in $T$, we denote by $[c, c']$ the set of corners of $T$ encountered when starting from $c$, going anticlockwise around $T$, and stopping at $c'$. The property (2.2) yields the following bound on oriented distances in $A$:
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+Figure 4: Example of the mapping from a pointed, rooted planar Eulerian triangulation to a well-labeled trees: top left, the original triangulation $A$, top right, the triangulation with the shifted labels. To determine the rooting of the associated tree $T$, we need an additional parameter $\epsilon \in \{0, 1\}$. Choosing it to be 0, we root $T$ at the edge remaining from the root face of $A$, with its original direction: bottom left, the tree with the labeling from top right, an bottom right, the shifted labels that respect the condition that the root corner has label 1. Here, the value of the parameter $\delta$ is 1, as the root edge of $A$ has type $3 \to 4$, and its root face has type 3.
+
+**Proposition 2.11.** Let $c, c'$ be two corners of $T$, with corresponding vertices $u, v$. Then
+
+$$ \vec{d}(u, v) \le 2 \left( l(u) + l(v) - 2 \min_{c'' \in [c, c')} l(c'') + 2 \right). $$
+
+*Proof.* Let $m = \min_{c'' \in [c, c]} l(c'')$, and let $c''$ be the first corner in $[c, c]$ such that $l(c'') = m$. Then $c''$ is the $(l(c)-m)$-th predecessor of $c$. Moreover, by definition, $p(c'')$ does not belong to $[c, c']$, so that it is also the $(l(c') - m + 1)$-th predecessor of $c'$. Thus, the predecessor geodesic $p(c'') \to c'' \to \dots \to c$, concatenated with the similar geodesic $p(c'') \to \dots \to c'$, is a simple path in $A$ made of $l(c) + l(c') - 2m + 2$ edges. However, part of it is not oriented from $c$ to $c'$, so that we lose a multiplicative factor of 2 when deducing a bound on the distance from $u$ to $v$. □
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+# HOMOLOGICAL STABILITY FOR THE RIBBON HIGMAN–THOMPSON GROUPS
+
+RACHEL SKIPPER AND XIAOLEI WU
+
+**ABSTRACT.** We generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman–Thompson groups, answering a question of Aramayona and Vlamis in the case of the Higman–Thompson groups. When the underlying surface is a disk, these new asymptotic mapping class groups can be identified with the ribbon and oriented ribbon Higman–Thompson groups. We use this model to prove that the ribbon Higman–Thompson groups satisfy homological stability, providing the first homological stability result for dense subgroups of big mapping class groups. Our result can also be treated as an extension of Szymik–Wahl’s work on homological stability for the Higman–Thompson groups to the surface setting.
+
+## INTRODUCTION
+
+The family of Thompson’s groups and the many groups in the extended Thompson family have long been studied for their many interesting properties. Thompson’s group $F$ is the first example of a type $F_\infty$, torsion-free group with infinite cohomological dimension [BG84] while Thompson’s groups $T$ and $V$ provided the first examples of finitely presented simple groups. More recently the braided and labeled braided Higman–Thompson groups have garnered attention in part due to their connections with big mapping class groups [Bro06, Deh06, AC20, SW]. In particular, Thumann constructed the ribbon version of Thompson’s group $V$ and proved that it is of type $F_\infty$ [Thu17]. The authors studied the ribbon Higman–Thompson groups $RV_{d,r}$ and their oriented version $RV_{d,r}^+$ in [SW]. In fact, we identified them with the so-called labeled braided Higman-Thompson groups and proved that they are all of type $F_\infty$.
+
+The homology of Thompson’s groups has also been an well-studied topic. Brown and Geoghegan computed the homology of $F$ in [BG84]; Ghys and Sergiescu calculated the homology of $T$ in [GS87]. More recently Szymik and Wahl showed that $V$ is acyclic [SW19], answering a question due to Brown [Bro92]. One of the key ingredients for their proof was showing that the Higman–Thompson groups $V_{d,1} \hookrightarrow V_{d,2} \hookrightarrow \dots \hookrightarrow V_{d,r} \hookrightarrow \dots$ satisfy homological stability for any fixed $d$. Recall that a family of groups $G_1 \hookrightarrow G_2 \hookrightarrow \dots \hookrightarrow G_n \hookrightarrow \dots$ is said to satisfy homological stability if the induced maps $H_i(G_n) \to H_i(G_{n+1})$ are isomorphisms for sufficiently large $n$. Classical examples of families of groups which satisfy homological stability include symmetric groups [Nak61], general linear groups [vdK80] and mapping class groups of surfaces [Har85].
+
+In the present paper, we extend Szymik and Wahl’s work to the class of ribbon Higman–Thompson groups. To accomplish this, we first build a geometric model for the ribbon Higman–Thompson groups using Funar–Kapoudjian’s asymptotic mapping class groups [FK04]. These groups are defined using a rigid structure on a surface minus a Cantor set and they sit naturally inside the ambient big mapping class groups. More recently, Aramayona and Funar [AF17] generalized the definition to surfaces with nonzero genus. In fact, Aramayona and Funar showed that the half-twist version of their asymptotic mapping class group (cf. Definition 3.14) is dense in the big mapping class group [AF17, Theorem 1.3]. Another surprising result of Funar and Neretin says that the half-twist asymptotic mapping class group of a closed surface minus a
+
+*Date:* May 2021.
+2010 Mathematics Subject Classification. 20F36, 57M07, 19D23, 20J05.
+Key words and phrases. Braided Higman–Thompson groups, ribbon Higman–Thompson groups, asymptotic mapping class groups, big mapping class groups, finiteness property, homological stability.
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+in the interior of the surface bounded by $\alpha$. Then the surface bounded by $\alpha$ and $A$ must be an annulus. From here one can produce an arc connecting the base point 0 to a point in $A$. Together with $A$, this provides the lollipop.
+
+Finally, we argue that $g$ is injective. Suppose $L_1$ and $L_2$ are two lollipops such that $g(L_1)$ and $g(L_2)$ are isotopic, denote the isotopy by $f$. By the isotopy extension theorem (see for example [FM11, Proposition 1.11]) there exists an isotopy $F: D_{d,r}^{\infty} \times [0, 1] \rightarrow D_{d,r}^{\infty}$ such that $F|_{D_{d,r}^{\infty} \times 0} = id_{D_{d,r}^{\infty}}$ and $F|_{g(L_1) \times [0,1]} = f$. In particular $F|_{D_{d,r}^{\infty} \times 1}$ maps the almost admissible loop $g(L_1)$ to the almost admissible loop $g(L_2)$. Hence $L_1$ is isotoped through $F$ to a lollipop which lies in a small neighborhood of $L_2$ and is bounded by the loop $g(L_2)$. Therefore, one can then isotope $L_1$ to $L_2$. $\square$
+
+We now have the following definition of lollipop complex.
+
+**Definition 4.21.** The lollipop complex $L_r^\infty(D, D_{d,1}^\infty)$ has vertices as lollipops, and $p+1$ lollipops, $L_0, L_1, \dots, L_p$, form a $p$-simplex if they are pairwise disjoint outside the base point 0 and there exists at least one admissible loop which does not lie inside the disks bounded by the $L_i$s.
+
+The following lemma is immediate from Lemma 4.20.
+
+**Lemma 4.22.** The complex $L_r^\infty(D, D_{d,1}^\infty)$ is isomorphic to $T_r^\infty(D, D_{d,1}^\infty)$ as a simplicial complex.
+
+**Lemma 4.23.** Given a $p$-simplex $\sigma$ in $L_r^\infty(D, D_{d,1}^\infty)$, its link $Lk(\sigma)$ is isomorphic to $L_{r_\sigma}^\infty(D, D_{d,1}^\infty)$ for some $r_\sigma > 0$.
+
+**Proof.** By Lemma 4.22, we can just prove the lemma for $T_r^\infty(D, D_{d,1}^\infty)$. Let $\alpha_0, \alpha_1, \dots, \alpha_p$ be the vertices of $\sigma$ which are almost admissible loops. Up to isotopy, we can assume they are pairwise disjoint except at the basepoint 0. Now let $C$ be the complement surface of $\sigma$, whose based boundary is the concatenation of $\alpha_p, \alpha_{p-1}, \dots, \alpha_0$ and $\partial D$. The surface $C$ has a naturally induced $d$-rigid structure. In particular, $C$ is asymptotically rigidly homeomorphic to $D_{d,r_\sigma}^\infty$ for some $r_\sigma > 0$. Thus link $Lk(\sigma)$ is isomorphic to $T_{r_\sigma}^\infty(D, D_{d,1}^\infty)$. $\square$
+
+Let us summarize the relationships we have so far between our various complexes by the following diagram.
+
+In Proposition 4.27, we will deduce the connectivity of $U_r^\infty(D, D_{d,1}^\infty)$ using the connectivity of the lollipop complex $L_r^\infty(D, D_{d,1}^\infty)$ by applying a bad simplices argument. Our goal now is to show that $L_r^\infty(D, D_{d,1}^\infty)$ is highly connected. Let us make some definitions first.
+
+**Definition 4.24.** Given any lollipop $L: (A, 0) \rightarrow (D_{d,r}^\infty, 0)$, we define the *free height* $\mathfrak{h}_L$ to be the minimal number $m$ such that $L([1, 2])$ is contained in $D_{d,r,m}$ up to free isotopy. We also define the height of an admissible loop to be the minimal number $m$ such that it is contained in $D_{d,r,m}$.
+
+To analyze the connectivity of $L_r^\infty(D, D_{d,1}^\infty)$, we need the following lemma which is a direct translation of [SW19, Lemma 3.8].
+
+**Lemma 4.25.** For any $r,p,N \ge 1$, there exists a number $\mathfrak{h}_{r,p,N} \ge 0$, such that for any $p$-simplex $\sigma$ in $L_r^\infty(D, D_{d,1}^\infty)$, and any $\mathfrak{h} \ge \mathfrak{h}_{r,p,N}$, there are at least $N$ lollipops of free height $\mathfrak{h}$ in $L_r^\infty(D, D_{d,1}^\infty)$ that are in $Lk(\sigma)$.
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+**Proof.** Note that for any vertex $L$ in $L_r^\infty(D, D_{d,1}^\infty)$, $L|_{[1,2]}$ is an admissible loop in $\oplus_r D_{d,1}^\infty$. Recall the function $q$ defined in Definition 3.16 which maps an admissible loop to an edge midpoint in the tree $\mathcal{T}_{d,r}$. Since each edge has a unique descendent vertex, we can instead map the loop to this vertex which lies in the forest $\mathcal{F}_{d,r}$. Using this connection, we can now choose $\mathfrak{h}_{r,p,N}$ to be the same as in [SW19, Lemma 3.8]. Then we have at least $N$ admissible loops of height $\mathfrak{h} \ge \mathfrak{h}_{r,p,N}$ which lie in the complement of the surface corresponding to $\sigma$ in $\oplus_r D_{d,1}^\infty$. Connecting each of these admissible loops to the base point in the complement surface, we get a set of lollipops in $Lk(\sigma)$. $\square$
+
+We now show that the complex $L_r^\infty(D, D_{d,1}^\infty)$ is in fact contractible. The idea of proof is similar to that of [SW19, Proposition 3.1] but with significantly more technical difficulty.
+
+**Proposition 4.26.** *The complex $L_r^\infty(D, D_{d,1}^\infty)$ is contractible for any $r \ge 1$.*
+
+**Proof.** The complex $L_r^\infty(D, D_{d,1}^\infty)$ is obviously non-empty. We will show by induction that for all $k \ge 0$, any map $S^k \to L_r^\infty(D, D_{d,1}^\infty)$ is null-homotopic. Assume $L_r^\infty(D, D_{d,1}^\infty)$ is $(k-1)$-connected.
+
+Let $f: S^k \to L_r^\infty(D, D_{d,1}^\infty)$ be a map. As usual, we can assume that the sphere $S^k$ comes with a triangulation such that the map $f$ is simplicial. We first use Lemma 1.7 to make the map $S^k$-simplexwise injective. For that we need for every $p$-simplex $\sigma$ in $L_r^\infty(D, D_{d,1}^\infty)$, its link $Lk(\sigma)$ is $(k-p-2)$-connected. But by Lemma 4.23, $Lk(\sigma)$ can be identified with $L_{r_\sigma}^\infty(D, D_{d,1}^\infty)$ for some $r_\sigma \ge 1$, so we have it is $(k-1)$-connected. Thus by Lemma 1.7, we can homotope $f$ to a map that is simplexwise injective.
+
+Now since $S^k$ is a finite simplicial complex, the free height of the vertices of $S^k$ has a maximum value. We first want to homotope $f$ to a new map such that all the vertices have free height at least $\mathfrak{h} = \mathfrak{h}_{r,k,N}$ where $N = v_0 + v_1 + \dots + v_k + 2$, where $v_i$ is the number of $i$-simplices of $S^k$ and $\mathfrak{h}_{r,k,N}$ is determined by Lemma 4.25. For that we use a bad simplices argument.
+
+We call a simplex of the sphere $S^k$ bad if all of its vertices are mapped to vertices in $L_r^\infty(D, D_{d,1}^\infty)$ that have free height less than $\mathfrak{h}$. We will modify $f$ by removing the bad simplices inductively starting by those of the highest dimension. Let $\sigma$ be a bad simplex of maximal dimension $p$ among all bad simplices. We will modify $f$ and the triangulation of $S^k$ in the star of $\sigma$ in a way that does not add any new bad simplices. In the process, we will increase the number of vertices by at most 1 in each step, and not at all if $\sigma$ is a vertex. This implies that, after doing this for all bad simplices, we will have increased the number of vertices of the triangulation of $S^k$ by at most $v_1 + \dots + v_k$. As $S^k$ originally had $v_0$ vertices, at the end of the process its new triangulation will have at most $v = v_0 + v_1 + \dots + v_k$ vertices. There are two cases.
+
+**Case 1:** $p=k$. If the bad simplex $\sigma$ is of the dimension $k$ of the sphere $S^k$, then its image $f(\sigma)$ has a complement loop which bounds a surface $C$ asymptotically rigidly homeomorphic to $D_{d,r_\sigma}^\infty$ for some $r_\sigma \ge 1$ by Lemma 4.23. Now we can choose a lollipop $y$ in $C$ with free height at least $\mathfrak{h}+1$. In particular $f(\sigma) \cup y$ form a $(k+1)$-simplex. We can then add a vertex $a$ in the center of $\sigma$, replacing $\sigma$ by $\partial\sigma * a$ and replacing $f$ by the map $(f|_{\partial\sigma}) * (a \mapsto y)$ on $\partial\sigma * a$. This map is homotopic to $f$ through the simplex $f(\sigma) \cup \{y\}$. We have added a single vertex to the triangulation. Because $L$ has free height $\mathfrak{h}+1$, we have not added any new bad simplices, and we have removed one bad simplex, namely $\sigma$. Moreover, $f$ remains simplexwise injective.
+
+**Case 2:** $p < k$. If the bad simplex $\sigma$ is a $p$-simplex for some $p < k$, by maximality of its dimension, the link of $\sigma$ is mapped to vertices of free height at least $\mathfrak{h}$ in the complement of the subsurface $f(\sigma)$. The simplex $\sigma$ has $p+1$ vertices whose images are pairwise disjoint outside the based point up to based isotopy. By Lemma 4.25 and our choice of $\mathfrak{h}$, there are at least $N = v+2$ lollipops $y_1, \dots, y_N$ of free height $\mathfrak{h}$ such that each $f(\sigma) \cup \{y_i\}$ form a $(p+1)$-simplex. As there are fewer vertices in the link than in the whole sphere $S^k$, and $S^k$ has at most $v$ vertices, by the pigeonhole principle, the loop part of the vertices in $f(Lk(\sigma))$ are contained in at most $v$ punctured disks bounded by the corresponding admissible loops with free height $\mathfrak{h}$. As $N = v+2$, there are at least two of the above vertices $y_i$ and $y_j$ of free height $\mathfrak{h}$ such that
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+standard ternary Cantor set is in fact isomorphic to its smooth mapping class group [FN18, Corollary 2]. In [AV20, Question 5.37], the following question was raised by Aramayona and Vlamis.
+
+**Question.** Are there other geometrically defined subgroups of $\text{Map}(\Sigma_g)$ which surject to other interesting classes of subgroups of homeomorphism group of the Cantor set, such as the Higman-Thompson groups, Neretin groups, etc?
+
+We proceed to construct two new classes of asymptotic mapping class groups, one of which answers their question in the case of Higman-Thompson groups while the other family surjects to the symmetric Higman-Thompson groups $V_{d,r}(\mathbb{Z}/2\mathbb{Z})$.
+
+**Theorem (3.17, 3.20).** Let $\Sigma$ be any compact surface and $C$ be a Cantor set which lies in the interior of a disk in $\Sigma$. Then the mapping class group $\text{Map}(\Sigma \setminus C)$ contains the following two families of dense subgroups: the asymptotic mapping class groups $\mathcal{B}V_{d,r}(\Sigma)$, which surject to the Higman-Thompson groups $V_{d,r}$, and the half-twist asymptotic mapping class groups $\mathcal{H}V_{d,r}(\Sigma)$, which surject to the symmetric Higman-Thompson groups $V_{d,r}(\mathbb{Z}/2\mathbb{Z})$.
+
+When $\Sigma$ is the disk, we identify $\mathcal{H}V_{d,r}(\Sigma)$ with the ribbon Higman-Thompson group $RV_{d,r}$ and $\mathcal{B}V_{d,r}(\Sigma)$ with the oriented ribbon Higman-Thompson group $RV_{d,r}^+$ (cf. Theorem 3.24). Using this geometric model for the ribbon Higman-Thompson groups, we are able to prove the following.
+
+**Theorem (4.30, 4.31).** Suppose $d \ge 2$. Then the inclusion maps induce isomorphisms
+
+$$ \iota_{R,d,r}: H_i(RV_{d,r}, M) \to H_i(RV_{d,r+1}, M) $$
+
+in homology in all dimensions $i \ge 0$, for all $r \ge 1$ and for all $H_1(RV_{d,\infty})$-modules $M$. The same also holds for the oriented ribbon Higman-Thompson groups $RV_{d,r}^+$.
+
+To the best of our knowledge, this is the first homological stability result for dense subgroups of big mapping class groups. Our proof uses a recent convenient framework given by Randal-Williams and Wahl [RWW17]. The core of the proof is similar to [SW19], but with new technical difficulties arising from infinite type surface topology. We hope our result here can be further used to calculate the homology of ribbon Higman-Thompson groups and shed light on the question whether braided $V$ is acyclic.
+
+**Outline of paper.** In Section 1, we describe the connectivity tools that will be necessary for the remainder of the paper. In Section 2, we introduce the definition of the Higman-Thompson, ribbon Higman-Thompson, and oriented ribbon Higman-Thompson groups using paired forest diagrams to define the elements. In Section 3, we generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman-Thompson groups. And finally, in Section 4, we prove homological stability for the ribbon Higman-Thompson groups and their oriented version.
+
+**Notation and convention.** All surfaces in this paper are assumed to be connected and orientable unless otherwise stated. Given a simplicial complex $X$ and a cell $\sigma \in X$, we denote the link of $\sigma$ in $X$ by $Lk_X(\sigma)$ (resp. the star of $\sigma$ by $\text{St}_X(\sigma)$). When the situation is clear, we quite often omit $X$ and simply denote the link by $Lk(\sigma)$ and the star by $\text{St}(\sigma)$. We also use the convention that $(-1)$-connected means non-empty and that every space is $(-2)$-connected. In particular, the empty set is $(-2)$-connected. Finally, we adopt the convention that elements in groups are multiplied from left to right.
+
+**Acknowledgements.** The first part of this project was done while the first author was a visitor in the Unité de mathématiques pures et appliquées at the ENS de Lyon and during a visit to the University of Bonn. She thanks them for their hospitality. She was also supported by the GIF, grant I-198-304.1-2015, “Geometric exponents of random walks and intermediate growth groups” and NSF DMS-2005297 “Group Actions on Trees and Boundaries of Trees”.
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+FIGURE 7. Replacing $\beta$ by $\beta'$ to reduce the number of intersection points with $y_i$.
+
+any loop parts of vertices in $f(\mathrm{Lk}(\sigma))$ are disjoint from the loop parts of $y_i$ and $y_j$. We can further assume that the arc parts of $y_i$ and $y_j$ never intersect with any loop part of the vertices in $f(\mathrm{Lk}(\sigma))$. And up to replacing the loop part of $y_i$ and $y_j$ with an admissible loop lying inside the disk bounded by the loop parts of $y_i$ and $y_j$ (note that this may increase the free height of $y_i$ and $y_j$), we can further assume that the arc parts of vertices in $f(\mathrm{Lk}(\sigma))$ are disjoint from the loop part of $y_i$ and $y_j$. But unlike the situation in the proof of [SW19, Proposition 3.1], a new problem we are facing here is that the arc parts of $y_i$ or $y_j$ might intersect the arc parts of the vertices in $f(\mathrm{Lk}(\sigma))$ even up to isotopy. In particular, given a simplex $\tau$ lying in the link of $\sigma$, $f(\sigma) \cup f(\tau) \cup y_i$ does not necessarily form a simplex now.
+
+For that we want to apply the mutual link trick (cf. Lemma 1.8) to remove the intersections of $f(\mathrm{Lk}(\sigma))$ with $y_i$ via a sequence of homotopies. In the process, we will only modify $f$ on $\mathrm{Lk}(\sigma)$ and the new map still maps $\mathrm{Lk}(\sigma)$ to $\mathrm{Lk}_{L_r^\infty(D,D_{d,1}^\infty)}(f(\sigma))$. Recall that $f$ is simplexwise injective. Up to isotopy, we can further choose representatives for vertices in $f(\mathrm{Lk}(\sigma))$ such that the intersection points of vertices in $f(\mathrm{Lk}(\sigma))$ and $y_i$ are isolated. Moreover, we assume the number of intersection points is minimal for each vertex in $f(\mathrm{Lk}(\sigma))$. Now we choose an intersection point $x_0$ in the arc $y_i([0, 1])$ that is closest to $y_i(1)$, denote the corresponding lollipop by $\beta$ which is the image of some vertex $b \in \mathrm{Lk}(\sigma)$. We can choose $\beta'$ to be a variation of $\beta$: $\beta'$ coincides with $\beta$ for the most part, except around the intersection point with $y_i$, we replace it by an arc going around the loop part of $y_i$. See Figure 7 for a picture of this. Now we apply Lemma 1.8, for which we need to check the following two conditions:
+
+(1) $f(\mathrm{Lk}_{S^k}(b)) \le \mathrm{Lk}_{L_r^\infty(D, D_{d,1}^\infty)}(\beta')$. This follows from our definition of $\beta'$. If a vertex $v$ in $f(\mathrm{Lk}_{S^k}(b))$ is disjoint from $\beta$, using the fact that the intersection point $x_0$ is the closest one to $y_i(1)$ and $f(v)$ is disjoint from the loop part of $y_i$, we have $\beta'$ is also disjoint from $v$.
+
+(2) $\mathrm{Lk}(\beta) \cap \mathrm{Lk}(\beta')$ is $(k-1)$-connected. The lollipops $\beta$ and $\beta'$ together will bound a disk which contains the loop part of $\beta$ and $y_i$. In any event, the complement of these is a surface asymptotically rigidly homeomorphic to some surface $D_{\alpha,\beta'}^\infty$ for some $r' \ge 1$. By our induction, it is $(k-1)$-connected.
+
+Now Lemma 1.8 says we can homotope $f$ to a new map such that $f(b) = \beta'$ and $f(\mathrm{Lk}(\sigma))$ has fewer intersection points with $y_i$. Step by step, at the end we have a simplexwise injective map $f$ such that for any vertex in $f(\mathrm{Lk}(\sigma))$, it only intersects with $y_i$ at the base point. In particular for any $\tau \in \mathrm{Lk}(\sigma)$, we have $f(\sigma) \cup f(\tau) \cup \{y_i\}$ forms a simplex in $L_r^\infty(D, D_{d,1}^\infty)$.
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+We can then replace $f$ inside the star
+
+$$ \mathrm{St}(\sigma) = \mathrm{Lk}(\sigma) * \sigma \simeq S^{k-p-1} * D^p $$
+
+by the map $(f|_{\mathrm{Lk}(\sigma)}) * (a \mapsto y_i) * (f|_{\partial\sigma})$ on
+
+$$ \mathrm{Lk}(\sigma) * a * \partial\sigma \simeq S^{k-p-1} * D^0 * S^{p-1}. $$
+
+which agrees with $f$ on the boundary $\mathrm{Lk}(\sigma) * \partial\sigma$ of the star, and is homotopic to $f$ through the map $(f|_{\mathrm{Lk}(\sigma)}) * (a \mapsto y_i) * (f|_{\sigma})$ defined on
+
+$$ \mathrm{Lk}(\sigma) * a * \sigma \simeq S^{k-p-1} * D^0 * D^p. $$
+
+Now $\mathrm{Lk}(\sigma) * a * \partial(\sigma)$ has exactly one extra vertex $a$ compared to the star of $\sigma$, unless $\sigma$ is just a vertex, in which case its boundary is empty and it has the same number of vertices. As $y_i$ has height at least $\hbar$, we have not added any new bad simplices. Hence we have reduced the number of bad simplices by one by removing $\sigma$.
+
+By induction, we can now assume that there are no bad simplices for $f$ with respect to a triangulation with at most $v$ vertices. With this assumption, we want to cone off $f$ just as we coned off the links in the above argument. We have more than $N = v + 2$ vertices of free height $\hbar$ in $L_r^\infty(D, D_{d,1}^\infty)$, and at most $v$ vertices in the sphere. The loop parts of these vertices are admissible loops of height at least $\hbar$. By the pigeonhole principle, we know that there are at least two lollipops $z_i$ and $z_j$ of free height $\hbar$ such that the punctured disks bounded by their loop parts are disjoint from the punctured disk bounded by any loop part of the lollipops in the vertices of $f(S^k)$. Just as before, we can further assume that the arc parts of $z_i$ and $z_j$ never intersect with any loop part of the vertices in $f(S^k)$, and the arc parts of vertices in $f(S^k)$ are disjoint from the loop part of $z_i$ and $z_j$. But the same problem appears again, as we want vertices of $f(S^k)$ to be disjoint from the whole lollipop $z_i$. For that we apply Lemma 1.8 again and the same proof as before implies that we can homotope $f$ such that its image is disjoint from $z_i$. In particular $f(S^k)$ lies in the link of $z_i$. Hence we can homotope $f$ to a constant map since $\mathrm{St}(z_i)$ is contractible. □
+
+**Proposition 4.27.** The complex $U_r^\infty(D, D_{d,1}^\infty)$ is contractible.
+
+**Proof.** As $T_r^\infty(D, D_{d,1}^\infty)$ is a subcomplex of $U_r^\infty(D, D_{d,1}^\infty)$, we can use a bad simplices argument.
+
+We call a vertex of $U_r^\infty(D, D_{d,1}^\infty)$ bad if it does not lie in $T_r^\infty(D, D_{d,1}^\infty)$ and a simplex bad if all of its vertices are bad. Given a bad $p$-simplex $\sigma$, we need to determine the connectivity of the good link $G_\sigma$ (see Subsection 1.2 for the definition of $G_\sigma$). As in the proof of Lemma 4.23, we have a complement surface $C_\sigma$ of $\sigma$ in $D_{d,1}^\infty$. Note that $C_\sigma$ inherits a $d$-rigid structure and it is asymptotically rigidly homeomorphic to $\oplus_{r_\sigma} D_{d,1}^\infty$ for some $r_\sigma > 0$. In particular, we can now identify $G_\sigma$ with $T_{r_\sigma}^\infty(D, D_{d,1}^\infty)$ which is contractible. Thus by Proposition 1.5, we have the pair $(U_r^\infty(D, D_{d,1}^\infty), T_r^\infty(D, D_{d,1}^\infty))$ is $i$-connected for any $i \ge 0$. By Proposition 4.26, $T_r^\infty(D, D_{d,1}^\infty) \cong L_r^\infty(D, D_{d,1}^\infty)$ is contractible, so we also have $U_r^\infty(D, D_{d,1}^\infty)$ is contractible. □
+
+**Corollary 4.28.** The complex $U_r(D, D_{d,1}^\infty)$ is weakly Cohen-Macaulay of dimension $r-2$.
+
+**Proof.** Note first that a simplicial complex is $(r-3)$-connected if and only if its $(r-2)$-skeleton is. Since $U_r(D, D_{d,1}^\infty)$ has the same $(r-2)$-skeleton as $U_r^\infty(D, D_{d,1}^\infty)$ and $U_r^\infty(D, D_{d,1}^\infty)$ is contractible, in particular $(r-3)$-connected, we indeed have $U_r(D, D_{d,1}^\infty)$ is $(r-3)$-connected.
+
+Now let $\sigma$ be a $p$-simplex of $U_r(D, D_{d,1}^\infty)$, with vertices $\phi_0, \phi_1, \dots, \phi_p$. We need to check that the link $\mathrm{Lk}_{U_r(D,D_{d,\sigma}^\infty)}(\sigma)$ is $(r-p-4)$-connected. We can assume $p \le r-3$ as any space is $(-2)$-connected. Moreover, it suffices to show the $(r-p-3)$-skeleton of $\mathrm{Lk}_{U_r(D,D_{d,\sigma}^\infty)}(\sigma)$ is $(r-p-4)$-connected. Since $\phi_0, \phi_1, \dots, \phi_p$ forms a $p$-simplex, similar to the proof of Lemma 4.23, we have the complement surface of $\sigma$ is asymptotically rigidly homeomorphic to some $d$-rigid surface $D_{d,\kappa_\sigma}^\infty$ for some $\kappa_\sigma > 0$. Then we can identify the $(r-p-3)$-skeleton of $\mathrm{Lk}_{U_r(D,D_{d,\sigma}^\infty)}(\sigma)$ with the $(r-p-3)$-skeleton of $U_{\kappa_\sigma}^\infty(D, D_{d,\sigma}^\infty)$. Since $U_{\kappa_\sigma}^\infty(D, D_{d,\sigma}^\infty)$ is even contractible, we have the connectivity bound we need. □
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+Now by Lemma 4.14 and Proposition 1.3, we have the following.
+
+**Corollary 4.29.** The complexes $S_r(D, D_{d,1}^\infty)$ and $W_r(D, D_{d,1}^\infty) \cdot$ are weakly Cohen-Macaulay of dimension $r-2$.
+
+4.4. **Homological stability.** We are finally ready to prove the homological stability result.
+
+**Theorem 4.30.** Suppose $d \ge 2$. Then the inclusion maps induce isomorphisms
+
+$$\iota_{R^+,d,r} : H_i(\mathbb{R}V_{d,r}^+, M) \to H_i(\mathbb{R}V_{d,r+1}^+, M)$$
+
+*in homology in all dimensions $i \ge 0$, for all $r \ge 1$ and for all $H_1(\mathbb{R}V_{d,\infty}^+)$-modules $M$.*
+
+**Proof.** By Theorem 4.6 and Corollary 4.29, choose $k=3$, we have for any abelian $\mathbb{R}V_\infty^+$-module $M$ the map
+
+$$H_i(\mathbb{R}V_r^+; M) \to H_i(\mathbb{R}V_{r+1}^+; M)$$
+
+induced by the natural inclusion map is surjective if $i \le \frac{r-k+2}{3}$, and injective if $i \le \frac{r-k}{3}$. But we can improve the stability range as in the proof of [SW19, Theorem 3.5] by noticing that we have the same canonical isomorphism between $\mathbb{R}V_{d,r}^+$ and $\mathbb{R}V_{d,r+d-1}^+$ using the ribbon braid model. This finishes the proof of Theorem 4.30. □
+
+**Theorem 4.31.** Suppose $d \ge 2$. Then the inclusion maps induce isomorphisms
+
+$$\iota_{R,d,r} : H_i(\mathbb{R}V_{d,r}, M) \to H_i(\mathbb{R}V_{d,r+1}, M)$$
+
+*in homology in all dimensions $i \ge 0$, for all $r \ge 1$ and for all $H_1(\mathbb{R}V_{d,\infty})$-modules $M$.*
+
+**Sketch of Proof.** The proof will be exactly the same as that of Theorem 4.30. Note first that by Theorem 3.24, it is the same as proving the half-twist asymptotic mapping class groups $\mathcal{H}V_{d,r}$ have homological stability. We define the braided monoidal category $\mathcal{G}'_d$ to be the category with objects $\oplus_r D_{d,1}^\infty$, $r \ge 0$, $\oplus$ as the operation, and $D$ as the 0 object. When $r=s$, we define the morphisms $\operatorname{Hom}(\oplus_r D_{d,1}^\infty, \oplus_s D_{d,1}^\infty) = \mathcal{H}V_{d,r}$ which can also be understood as the group of isotopy classes of asymptotically quasi-rigid homeomorphisms of $\oplus_r D_{d,1}^\infty$; when $r \ne s$, let $\operatorname{Hom}(\oplus_r D_{d,1}^\infty, \oplus_s D_{d,1}^\infty) = \emptyset$. We then have a homogeneous category $\mathcal{U}\mathcal{G}'_d$ and to prove the homological stability for the sequence of groups $\mathcal{H}V_{d,1} \le \mathcal{H}V_{d,2} \le \dots$, we only need to prove the associated space $W_r(D, D_{d,1}^\infty) \cdot$, in fact the associated simplicial complex $S_r(D, D_{d,1}^\infty)$, is highly connected. At this point, the complex is slightly different from the oriented case, but still the new complex $S_r(D, D_{d,1}^\infty)$ is a complete join over the old complex $U_r(D, D_{d,1}^\infty)$. Hence, the connectivity of $S_r(D, D_{d,1}^\infty)$ again follows from Corollary 4.28 and Proposition 1.3. □
+
+## REFERENCES
+
+[AC20] Julio Aroca and María Cumplido. A new family of infinitely braided Thompson's groups. *J. Algebra*, 2020. To appear. arxiv: 2005.09593.
+
+[AF17] Javier Aramayona and Louis Funar. Asymptotic mapping class groups of closed surfaces punctured along Cantor sets. *Moscow Math. J.*, 2017. to appear. arXiv:1701.08132.
+
+[AV20] Javier Aramayona and Nicholas G. Vlamis. Big mapping class groups: an overview. In Ken'ichi Ohshika and Athanase Papadopoulos, editors, *In the Tradition of Thurston: Geometry and Topology*, pages 459–496. Springer, 2020.
+
+[BDJ17] Collin Bleak, Casey Donoven, and Julius Jonušas. Some isomorphism results for Thompson-like groups $V_n(G)$. *Israel J. Math.*, 222(1):1–19, 2017.
+
+[BFM$^+$16] Kai-Uwe Bux, Martin G. Fluch, Marco Marschler, Stefan Witzel, and Matthew C. B. Zaremsky. The braided Thompson’s groups are of type $F_\infty$. *J. Reine Angew. Math.*, 718:59–101, 2016. With an appendix by Zaremsky.
+
+[BG84] Kenneth S. Brown and Ross Geoghegan. An infinite-dimensional torsion-free $FP_\infty$ group. *Invent. Math.*, 77(2):367–381, 1984.
+
+[Bro92] Kenneth S. Brown. The geometry of finitely presented infinite simple groups. In *Algorithms and classification in combinatorial group theory* (Berkeley, CA, 1989), volume 23 of *Math. Sci. Res. Inst. Publ.*, pages 121–136. Springer, New York, 1992.
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+We can apply the proposition in the following way.
+
+**Theorem 1.6.** [HV17, Corollary 2.2] Let $Y$ be a subcomplex of a simplicial complex $X$ and suppose the space $X \setminus Y$ has a set of bad simplices satisfying (1) and (2) above, then:
+
+(1) If $X$ is $n$-connected and $G_\sigma$ is $(n - \dim(\sigma))$-connected for all bad simplices $\sigma$, then $Y$ is $n$-connected.
+
+(2) If $Y$ is $n$-connected and $G_\sigma$ is $(n - \dim(\sigma) - 1)$-connected for all bad simplices $\sigma$, then $X$ is $n$-connected.
+
+1.3. **The mutual link trick.** In the proof of [BFM$^+$16, Theorem 3.10], there is a beautiful argument for resolving intersections of arcs inspired by Hatcher's flow argument [Hat91]. They attributed the idea to Andrew Putman. Recall Hatcher's flow argument allows one to "flow" a complex to its subcomplex. But in the process, one can only "flow" a vertex to a new one in its link. The mutual link trick will allow one to "flow" a vertex to a new one not in its link provided "the mutual link" is sufficiently connected.
+
+To apply the mutual link trick, we first need a lemma that allows us to homotope a simplicial map to a simplexwise injective one [BFM$^+$16, Lemma 3.9]. Recall a simplicial map is called simplexwise injective if its restriction to any simplex is injective. See also [GRW18, Section 2.1] for more information.
+
+**Lemma 1.7.** Let $Y$ be a compact $m$-dimensional combinatorial manifold. Let $X$ be a simplicial complex and assume that the link of every $p$-simplex in $X$ is $(m-p-2)$-connected. Let $\psi: Y \to X$ be a simplicial map whose restriction to $\partial Y$ is simplexwise injective. Then after possibly subdividing the simplicial structure of $Y$, $\psi$ is homotopic relative $\partial Y$ to a simplexwise injective map.
+
+Note that as discussed in [GLU20, Lemma 5.19], there is a mistake in the connectivity bound given in [BFM$^+$16] that has been corrected here.
+
+**Lemma 1.8 (The mutual link trick).** Let $Y$ be a closed $m$-dimensional combinatorial manifold and $f: Y \to X$ be a simplexwise injective simplicial map. Let $y \in Y$ be a vertex and $f(y) = x$ for some $x \in X$. Suppose $x'$ is another vertex of $X$ satisfying the following condition.
+
+(1) $f(\mathrm{Lk}_Y(y)) \leq \mathrm{Lk}_X(x')$,
+
+(2) the mutual link $\mathrm{Lk}_X(x) \cap \mathrm{Lk}_X(x')$ is $(m-1)$-connected,
+
+Then we can define a new simplexwise injective map $g: Y \to X$ by sending $y$ to $x'$ and all the other vertices $y'$ to $f(y')$ such that $g$ is homotopic to $f$.
+
+**Proof.** The conditions that $f$ is simplexwise injective and $f(\mathrm{Lk}_Y(y)) \leq \mathrm{Lk}_X(x')$ guarantee that the definition of $g$ can be extended over $Y$ and $g$ is again simplexwise injective.
+
+We need to prove $g$ is homotopic to $f$. The homotopy will be the identity outside $\mathrm{St}_Y(y)$. Note that since $f$ is simplexwise injective, we have $f(\mathrm{Lk}_Y(y)) \leq \mathrm{Lk}_X(x)$. Together with Condition (1), we have $f(\mathrm{Lk}_Y(y)) \leq \mathrm{Lk}_X(x) \cap \mathrm{Lk}_X(x')$. Since $\mathrm{Lk}_Y(y)$ is an $(m-1)$-sphere and $\mathrm{Lk}_X(x) \cap \mathrm{Lk}_X(x')$ is $(m-1)$-connected, there exists an $m$-disk $B$ with $\partial B = \mathrm{Lk}_Y(y)$ and a simplicial map $\varphi: B \to \mathrm{Lk}_X(x) \cap \mathrm{Lk}_X(x')$ so that $\varphi$ restricted to $\partial B$ coincides with $\psi$ restricted to $\mathrm{Lk}_Y(y)$. Since the image of $B$ under $\varphi$ is contained in $\mathrm{St}_X(x)$ which is contractible, we can homotope $g$, replacing $g|_{\mathrm{St}_Y(y)}$ with $\varphi$. Since the image of $B$ under $f$ is also contained in $\mathrm{Lk}_X(x')$, we can similarly homotope $f$, replacing $f|_{\mathrm{St}_Y(y)}$ with $\varphi$. These both yield the same map, so $g$ is homotopic to $f$. $\square$
+
+## 2. HIGMAN–THOMPSON GROUPS AND THEIR BRAIDED VERSIONS
+
+In this section, we first give an introduction to the Higman–Thompson groups and then introduce their ribbon version.
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+2.1. **Higman–Thompson groups.** The Higman–Thompson groups were first introduced by Higman as a generalization of the groups [Hig74] given earlier in handwritten, unpublished notes of Richard Thompson. First let us recall the definition of the Higman–Thompson groups. Although there are a number of equivalent definitions of these groups, we will use the notion of paired forest diagrams. First we define a *finite rooted d-ary tree* to be a finite tree such that every vertex has degree $d+1$ except the *leaves* which have degree 1, and the *root*, which has degree $d$ (or degree 1 if the root is also a leaf). Usually we draw such trees with the root at the top and the nodes descending from it down to the leaves. A vertex $v$ of the tree along with its $d$ adjacent descendants will be called a *caret*. If the leaves of a caret in the tree are leaves of the tree, we will call the caret *elementary*. A collection of $r$ many $d$-ary trees will be called a $(d,r)$-forest. When $d$ is clear from the context, we may just call it an $r$-forest.
+
+Define a *paired $(d,r)$-forest diagram* to be a triple $(F_-, \rho, F_+)$ consisting of two $(d,r)$-forests $F_-$ and $F_+$ both with $l$ leaves for some $l$, and a permutation $\rho \in S_l$, the symmetric group on $l$ elements. We label the leaves of $F_-$ with $1, \dots, l$ from left to right, and for each $i$, the $\rho(i)^{\text{th}}$ leaf of $F_+$ is labeled $i$.
+
+Define a *reduction* of a paired $(d, r)$-forest diagram to be the following: Suppose there is an elementary caret in $F_-$ with leaves labeled by $i, \dots, i+d-1$ from left to right, and an elementary caret in $F_+$ with leaves labeled by $i, \dots, i+d-1$ from left to right. Then we can “reduce” the diagram by removing those carets, renumbering the leaves and replacing $\rho$ with the permutation $\rho' \in S_{l-d+1}$ that sends the new leaf of $F_-$ to the new leaf of $F_+$, and otherwise behaves like $\rho$. The resulting paired forest diagram $(F'_-, \rho', F'_+)$ is then said to be obtained by *reducing* $(F_-, \rho, F_+)$. See Figure 1 below for an idea of reduction of paired (3,2)-forest diagrams. The reverse operation to reduction is called *expansion*, so $(F_-, \rho, F_+)$ is an expansion of $(F'_-, \rho', F'_+)$. A paired forest diagram is called *reduced* if there is no reduction possible. Define an equivalence relation on the set of paired $(d, r)$-forest diagrams by declaring two paired forest diagrams to be equivalent if one can be reached by the other through a finite series of reductions and expansions. Thus an equivalence class of paired forest diagrams consists of all diagrams having a common reduced representative. Such reduced representatives are unique.
+
+FIGURE 1. Reduction, of the top paired (3,2)-forest diagram to the bottom one.
+
+There is a binary operation $\ast$ on the set of equivalence classes of paired $(d, r)$-forest diagrams. Let $\alpha = (F-, \rho, F_+)$ and $\beta = (E-, \xi, E_+)$ be reduced paired forest diagrams. By applying repeated expansions to $\alpha$ and $\beta$ we can find representatives $(F'_-, \rho', F'_+)$ and $(E'_-, \xi', E'_+)$ of the equivalence classes of $\alpha$ and $\beta$, respectively, such that $F'_+ = E'_-$. Then we declare $\alpha \ast \beta$ to be $(F'_-, \rho'\xi', E'_+)$. This operation is well defined on the equivalence classes and is a group operation.
+
+**Definition 2.1.** The Higman–Thompson group $V_{d,r}$ is the group of equivalence classes of paired $(d, r)$-forest diagrams with the multiplication $\ast$.
+
+The usual Thompson group $V$ is a special case of Higman–Thompson groups. In fact, $V = V_{2,1}$.
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+(3) If an admissible boundary component $L$ of $A$ is contained in $\Sigma'$, then the punctured disk component of $\Sigma_{d,r}^\infty$ cutting along $L$ is also contained in $\Sigma'$.
+
+Then $\Sigma'$ has a naturally induced $d$-rigid structure. In fact, we can take $A \cap \Sigma'$ to be the base surface and $d$-rigid structure can simply be inherited from $\Sigma_{d,r}^\infty$. We, of course, can choose different $A$ here which may give different induced $d$-rigid structure, but it is unique up to asymptotically rigid homeomorphism.
+
+**3.2. Asymptotic mapping class groups surjecting to Higman-Thompson groups.**
+
+Given a (possibly noncompact) surface $\Sigma$, recall the mapping class group of $\Sigma$ is defined to be the group of isotopy classes of orientation preserving homeomorphisms of $\Sigma$ that fixes $\partial\Sigma$ pointwise, i.e.
+
+$$ \mathrm{Map}(\Sigma) = \mathrm{Map}(\Sigma, \partial\Sigma) := \mathrm{Homeo}^+(\Sigma, \partial\Sigma)/\mathrm{Homeo}_0(\Sigma, \partial\Sigma). $$
+
+With this, we can now define the asymptotic mapping class group and the half-twist asymptotic mapping class group.
+
+**Definition 3.14.** The asymptotic mapping class group $\mathcal{B}V_{d,r}(\Sigma)$ (resp. the half-twist asymptotic mapping class group $\mathcal{H}V_{d,r}(\Sigma)$) is the subgroup of $\mathrm{Map}(\Sigma_{d,r}^\infty)$ consisting of isotopy classes of asymptotically rigid (resp. quasi-rigid) self-homeomorphisms of $\Sigma_{d,r}^\infty$. When $\Sigma$ is the disk, we sometimes simply denote the group by $\mathcal{B}V_{d,r}$ (resp. $\mathcal{H}V_{d,r}$).
+
+**Definition 3.15.** Let $A$ be an admissible subsurface of $\Sigma_{d,r}^\infty$, and $\mathrm{Map}(A)$ be its mapping class group which fixes the each boundary component pointwise. Each inclusion $A \subseteq A'$ of admissible surfaces induces an injective embedding $j_{A,A'} : \mathrm{Map}(A) \to \mathrm{Map}(A')$. The collection forms a direct system whose direct limit we call the *compactly supported pure mapping class group*, denoted by $\mathrm{PMap}_c(\Sigma_{d,r}^\infty)$. The group $\mathrm{PMap}_c(\Sigma_{d,r}^\infty)$ is naturally a subgroup of $\mathcal{B}V_{d,r}(\Sigma)$ and we denote the inclusion map by $j$.
+
+**Definition 3.16.** Let $\mathcal{F}_{d,r}$ be the forest with $r$ copies of a rooted $d$-ary tree and $\mathcal{T}_{d,r}$ be the rooted tree obtained from $\mathcal{F}_{d,r}$ by adding an extra vertex to $\mathcal{F}_{d,r}$ and $r$ extra edges each connecting this vertex to a root of a tree in $\mathcal{F}_{d,r}$. There is a natural projection $q: \Sigma_{d,r}^\infty \to \mathcal{T}_{d,r}$, such that the pullback of the root is $\Sigma_{d,r,0}$ and the pull back of the midpoints of any edges are admissible loops.
+
+Now any element in $\mathcal{B}V_{d,r}(\Sigma)$ can be represented by an asymptotically rigid homeomorphism $\varphi: \Sigma_{d,r}^\infty \to \Sigma_{d,r}^\infty$. In particular we have an admissible subsurface $A$ of $\Sigma_{d,r}^\infty$ such that $\varphi|_A: (A, \partial_b A) \to (\varphi(A), \varphi(\partial_b A))$ is a homeomorphism. Let $F_-$ be the smallest subforest of $\mathcal{F}_{d,r}$ which contains $q(A) \cap \mathcal{F}_{d,r}$, and $F_+$ be the smallest subforest of $\mathcal{F}_{d,r}$ which contains $q(\varphi(A)) \cap \mathcal{F}_{d,r}$. Note that $F_-$ and $F_+$ have the same number of leaves and their leaves are in one-to-one correspondence with the admissible loops of $A$ and $\varphi(A)$. Now let $\rho$ be the map from leaves of $F_-$ to $F_+$ induced by $\varphi$. Together this defines an element $[(F_-, \rho, F_+)] \in V_{d,r}$. We call this map $\pi$. One can show $\pi$ is well defined. Similarly to [FK04, Proposition 2.4] and [AF17, Proposition 4.2, 4.6], we now have the following proposition.
+
+**Proposition 3.17.** We have the short exact sequences:
+
+$$ 1 \to \mathrm{PMap}_c(\Sigma_{d,r}^\infty) \xrightarrow{j} \mathcal{B}V_{d,r}(\Sigma) \xrightarrow{\pi} V_{d,r} \to 1; $$
+
+$$ 1 \to \mathrm{PMap}_c(\Sigma_{d,r}^\infty) \xrightarrow{j} \mathcal{H}V_{d,r}(\Sigma) \xrightarrow{\pi} V_{d,r}(\mathbb{Z}/2\mathbb{Z}) \to 1. $$
+
+**Remark 3.18.** Here, as in [AF17], $V_{d,r}(\mathbb{Z}/2\mathbb{Z})$ is the twisted version of the Higman-Thompson group where one allows flipping the subtree below every leaf. See for example [BDJ17] for more information.
+
+**Proof.** We will prove the proposition for $\mathcal{B}V_{d,r}(\Sigma)$. The other case is essentially the same. First we show the map $\pi$ is surjective. Given any element $[(F_-, \rho, F_+)] \in V_{d,r}$, let $T_-$ (resp. $T_+$) be the tree obtained from $F_-$ (resp. $F_+$) by adding a single root on the top and $r$ edges
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+2.2. **Ribbon Higman–Thompson groups.** For convenience, we will think of the forest $F_+$ drawn beneath $F_-$ and upside down, i.e., with the root at the bottom and the leaves at the top. The permutation $\rho$ is then indicated by arrows pointing from the leaves of $F_-$ to the corresponding paired leaves of $F_+$. See Figure 2 for this visualization of (the unreduced representation of) the element of $V_{3,2}$ from Figure 1.
+
+FIGURE 2. An element of $V_{3,2}$.
+
+Now in the ribbon version of the Higman–Thompson groups, the permutations of leaves are simply replaced by ribbon braids which can twist between the leaves.
+
+**Definition 2.2.** Let $\mathcal{I} = \Pi_{i=1}^d I_i : [0,1] \times \{1, \dots, l\} \to \mathbb{R}^2$ be an embedding which we refer to as the *marked bands*. A *ribbon braid* is a map $R : ([0,1] \times \{0,1,\dots,l\}) \times [0,1] \to \mathbb{R}^2$ such that for any $0 \le t \le 1$, $R_t : [0,1] \times \{1,\dots,l\} \to \mathbb{R}^2$ is an embedding, $R_0 = \mathcal{I}$, and there exists $\sigma \in S_l$ such that $R_1(t)|_{I_i} = I_{\sigma(i)}(t)$ or $R_1(t)|_{I_i} = I_{\sigma(i)}(1-t)$. The usual product of paths defines a group structure on the set of ribbon braids up to homotopy among ribbon braids. This group, denoted by $RB_l$, does not depend on the choice of the marked bands and it is called the ribbon braid group with $l$ bands.
+
+**Remark 2.3.** Note that $RB_l \cong \mathbb{Z}^l \ltimes B_l$ where the action of $B_l$ is induced by the symmetric group action on the coordinates of $\mathbb{Z}^l$.
+
+**Definition 2.4.** A *ribbon braided paired (d,r)-forest diagram* is a triple $(F_-, r, F_+)$ consisting of two $(d,r)$-forests $F_-$ and $F_+$ both with $l$ leaves for some $l$ and a ribbon braid $r \in RB_l$ connecting the leaves of $F_-$ to the leaves of $F_+$.
+
+The expansion and reduction rules for the ribbon braids just come from the natural way of splitting a ribbon band into $d$ components and the inverse operation to this. See Figure 3 for how to split a half twisted band when $d=2$. Note that not only are the two bands themselves twisted but the bands are also braided. Everything else will be the same as in the braided case, so we omit the details here. As usual, we define two ribbon braided paired forest diagrams to be equivalent if one is obtained from the other by a sequence of reductions or expansions. The multiplication operation $\ast$ on the equivalence classes is defined the same way as for $bV_{d,r}$. We direct the reader to [SW, Section 2]
+
+FIGURE 3. Splitting a ribbon into 2 ribbons.
+
+**Definition 2.5.** The *ribbon Higman–Thompson group* $RV_{d,r}$ (resp. the *oriented ribbon Higman–Thompson group* $RV_{d,r}^+$) is the group of equivalence classes of (resp. oriented) ribbon braided paired $(d,r)$-forests diagrams with the multiplication $\ast$.
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+### 3. ASYMPTOTIC MAPPING CLASS GROUPS RELATED TO THE RIBBON HIGMAN–THOMPSON GROUPS
+
+The purpose of this section is to generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman–Thompson groups. In particular, we will build a geometric model for the ribbon Higman–Thompson groups which will be crucial for proving homological stability in Section 4. Our construction is largely based on the ideas in [FK04, Section 2] and [AF17, Section 3].
+
+**3.1. d-rigid structure.** In this subsection, we generalize the notion of a rigid structure to that of a *d*-rigid structure.
+
+**Definition 3.1.** A $d$-leg pants is a surface which is homeomorphic to a $(d+1)$-holed sphere.
+
+Recall that the usual pair of pants is a 2-leg pants. We will draw a $d$-leg pants with one boundary component at the top. In this way, we can conveniently put a counter-clockwise total order on the boundary components, making the top component the minimal one. See Figure 4 for an example of a 3-leg pants.
+
+We proceed to build some infinite type surfaces using some basic building blocks.
+
+**Definition 3.2.** Let $\Sigma$ be an compact oriented surface. Call the boundary components of $\Sigma$ the *based boundary components*. Then $\Sigma_{d,r}^\infty$ is the infinite surface, built up as an inductive limit of infinite surfaces $\Sigma_{d,r,m}$ with $m \ge 0$:
+
+(1) $\Sigma_{d,r,0}$ is obtained from $\Sigma$ by deleting the interior of a disk in $\Sigma$. When $\Sigma$ is a disk $D$, we declare $D_{d,r,0} = \partial D$.
+
+(2) $\Sigma_{d,r,1}$ is obtained from $\Sigma_{d,r,0}$ attaching a copy of $r$-leg pants along the newly created boundary of $\Sigma_{d,r,0}$.
+
+(3) For $m \ge 1$, $\Sigma_{d,r,m+1}$ is obtained from $\Sigma_{d,r,m}$ by gluing a pair of $d$-leg pants to every nonbased boundary circle of $\Sigma_{d,r,m}$ along the top boundary of the pants.
+
+The surface $\Sigma_{d,r,1}$ is called the *base* of $\Sigma_{d,r}^\infty$ and the boundary components of $\Sigma_{d,r}^\infty$ coming from the base are the *based boundary components*. For each $m \ge 1$, the nonbased boundary components of $\Sigma_{d,r,m}$ naturally embed in $\Sigma_{d,r}^\infty$ and we call these the *admissible loops*. We call the admissible loops coming from $\Sigma_{d,r,1}$ the *rooted loops*. The surface $\Sigma_{d,r}^\infty$ has a natural induced orientation.
+
+**Remark 3.3.** To define our $d$-rigid structure, we do not really need $\Sigma_{d,r,0}$. But it will be convenient to have $\Sigma_{d,r,0}$ later in Definition 3.16 for defining the map from $\Sigma_{d,r}^\infty$ to the tree $T_{d,r}$.
+
+**Remark 3.4.** In the special case where the starting surface is a disk, we will use the notation $\Sigma = D$, $\Sigma_{d,r,m} = D_{d,r,m}$, and $\Sigma_{d,r}^\infty = D_{d,r}^\infty$. See Figure 4 for a picture of the surface $D_{3,3}^\infty$. In this case, we can think of $D_{d,r}^\infty$ as a subsurface of a disk $D$. More specifically, let $D = \{(x,y) | x^2 + y^2 \le 1\}$ and $x_i = \frac{2i-r-1}{r+1}$, $1 \le i \le r$. We place $r$ disks with center at each $(x_i, 0)$ of radius $r_0 = \frac{1}{4(r+1)}$. Denote these disks by $D_1, \cdots D_r$. The complement of the interior of these $r$ disks in $D$ is homeomorphic to the $r$-leg pants $D_{d,r,1}$. Now for each disk $D_i$, $1 \le i \le r$, we can equally distribute $d$ points in the $x$-axis inside $D_i$ and place a disk with radius $\frac{r_0}{d}$ centered at each. We have the complement of the interior of these $d$ disks in $D_i$ are all $d$-leg pants. We can continue the process inductively. At the end, the disks converge to a Cantor set which we denote by $C$. In particular $D_{d,r}^\infty$ is homeomorphic to $D \setminus C$. We will refer to this as the *puncture model* for $D_{3,3}^\infty$. See Figure 5 for a picture of $D_{3,3}^\infty$ with this model. The advantage of this model is we can view $D_{d,r}^\infty$ and all its admissible subsurfaces directly as a subsurfaces of $D$.
+
+**Remark 3.5.** Now $\Sigma_{d,r}^\infty$ can be obtained from $\Sigma$ by attaching a copy of $D_{d,r}^\infty$ to the nonbased boundary component of $\Sigma_{d,r,0}$. In particular, $\Sigma_{d,r}^\infty$ is obtained from $\Sigma$ by deleting a copy of the Cantor set, and any admissible subsurface of $\Sigma_{d,r}^\infty$ can be viewed directly as a subsurface of $\Sigma$ using the puncture model. Recall that any two Cantor sets are homeomorphic, hence, by the classification of infinite surfaces [AV20, Theorem 2.2], we have $\Sigma_{d,r}^\infty$ is homeomorphic to $\Sigma \setminus C$
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+FIGURE 4. 3-leg pants and the surface $D_{3,3}^{\infty}$ with canonical seams
+
+FIGURE 5. Disk model for the surface $D_{3,3}^{\infty}$
+
+where $\mathcal{C}$ is the standard ternary Cantor set sitting inside some disk in $\Sigma$ regardless of the choice of $d$ and $r$.
+
+**Definition 3.6.** A compact subsurface $A \subset \Sigma_{d,r}^{\infty}$ is *admissible* if $\Sigma_{d,r,1} \subseteq A$ and all of its nonbased boundaries are admissible. The subsurfaces $\Sigma_{d,r,m}$ are called the *standard admissible subsurfaces* of $\Sigma_{d,r}^{\infty}$.
+
+**Definition 3.7.**
+
+(1) A *suited d-pants decomposition* of the infinite surface $\Sigma_{d,r}^{\infty}$ is a maximal collection of distinct nontrivial simple closed curves in the interior of $\Sigma_{d,r}^{\infty} \setminus \Sigma_{d,r,1}$ which are not isotopic to the boundary, pairwise disjoint and pairwise non-isotopic, with the additional property that the complementary regions in $\Sigma_{d,r}^{\infty} \setminus \Sigma_{d,r,1}$ are all $d$-leg pants.
+
+(2) A *d-rigid structure* on $\Sigma_{d,r}^{\infty}$ consists of two pieces of data:
+
+• a suited *d*-pants decomposition, and
+
+• a *d-prerigid structure*, i.e. a countable collection of disjoint line segments embedded into $\Sigma_{d,r}^{\infty} \setminus \Sigma_{d,r,1}$, such that the complement of their union in each component of $\Sigma_{d,r}^{\infty} \setminus \Sigma_{d,r,1}$ has 2 connected components.
+
+These pieces must be *compatible* in the following sense: first, the traces of the *d*-prerigid structure on each *d*-leg pants (i.e. the intersections with pants) are made up of $d+1$
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+connected components, called *seams*; secondly, each boundary component of the pants intersects with exactly two components of the seams at two distinct points; thirdly, the seams cut each pants into two components. Note that these conditions imply that each component is homeomorphic to a disk. One says then that the suited *d*-pants decomposition and the *d*-prerigid structure are *subordinate* to the *d*-rigid structure.
+
+(3) By construction, $\Sigma_{d,r}^\infty$ is naturally equipped with a suited $d$-pants decomposition, which will be referred to below as the *canonical suited d-pants decomposition*. We also fix a $d$-prerigid structure on $\Sigma_{d,r}^\infty$ (called the *canonical d-prerigid structure*) compatible with the canonical suited $d$-pants decomposition. See Figure 4. Using the puncture model, the seams of the canonical $d$-prerigid structure are just the intersections of $[0,1] \times \{0\}$ with each $d$-pants. The resulting $d$-rigid structure is called the *canonical $d$-rigid structure* on $\Sigma_{d,r}^\infty$. Very importantly, for each admissible subsurface, the canonical $d$-rigid structure induces an order on the admissible boundaries. In Figure 4, the induced order on the admissible loops are counterclockwise. Using the puncture model, the admissible loops are ordered from left to right.
+
+(4) The seams cut each component of $\Sigma_{d,r}^\infty \setminus \Sigma_{d,r,1}$ into two pieces, we choose the front piece in each component and these $r$ pieces together form the *visible side* of $\Sigma_{d,r}^\infty$.
+
+(5) A suited $d$-pants decomposition (resp. $d$-(pre)rigid structure) is *asymptotically trivial* if outside a compact subsurface of $\Sigma_{d,r}^\infty$, it coincides with the canonical suited $d$-pants decomposition (resp. canonical $d$-(pre)rigid structure).
+
+**Remark 3.8.** It is important that the seams cut each $d$-pants into two components and each component is homeomorphic to a disk as the mapping class group of a disk is trivial.
+
+**Definition 3.9.** Let $\Sigma_{d,r}^\infty$ and $\bar{\Sigma}_{d,r'}^\infty$ be two surfaces with $d$-rigid structure and let $\varphi : \Sigma_{d,r}^\infty \to \bar{\Sigma}_{d,r'}^\infty$ be a homeomorphism. One says that $\varphi$ is *asymptotically rigid* if there exists an admissible subsurface $A \subset \Sigma_{d,r}^\infty$ such that:
+
+(1) $\varphi(A)$ is also admissible in $\bar{\Sigma}_{d,r'}^\infty$,
+
+(2) $\varphi|_A$ maps the based boundaries to based boundaries, admissible loops to admissible loops and
+
+(3) the restriction of $\varphi: \Sigma_{d,r}^\infty \setminus A \to \bar{\Sigma}_{d,r'}^\infty \setminus \varphi(A)$ is *rigid*, meaning that it respects the traces of the canonical $d$-rigid structure, mapping the suited $d$-pants decomposition into the suited $d$-pants decomposition, the seams into the seams, and the visible side into the visible side.
+
+If we drop the condition that $\varphi$ should map the visible side into the visible side, $\varphi$ is called *asymptotically quasi-rigid*. The surface $A$ is called a *support* for $\varphi$.
+
+**Remark 3.10.** We are not using the word “support” in the usual sense, as the map outside the support defined above might well not being the identity, but the map is uniquely determined up to isotopy by Remark 3.8.
+
+**Remark 3.11.** In [FK04, Definition 2.3], they do not actually require that the support must contain the base. This will not make a difference, as one can always enlarge the support so that it contains the base.
+
+**Remark 3.12.** The surface $\Sigma_{d,r+d-1}^\infty$ can be identified with the surface $\Sigma_{d,r}^\infty$ such that $\Sigma_{d,r+d-1,m} = \Sigma_{d,r,m+1}$ for any $m \ge 1$, and the $d$-rigid structure of $\Sigma_{d,r}^\infty$ coincides with $d$-rigid on $\Sigma_{d,r+d-1}^\infty$ outside $\Sigma_{d,r,2}$. In this way, $\Sigma_{d,r}^\infty$ is asymptotically rigid homeomorphic to $\Sigma_{d,r+d-1}^\infty$ through the identity map.
+
+**Remark 3.13.** Let $\Sigma'$ be a subsurface of $\Sigma_{d,r}^\infty$ such that there exist an admissible subsurface $A$ of $\Sigma_{d,r}^\infty$ satisfying:
+
+(1) $A \cap \Sigma'$ is a compact surface,
+
+(2) The boundaries of $\Sigma'$ are disjoint from the admissible boundary components of $A$.
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+connecting to each root of the trees in $F_-$ (resp. $F_+$). Furthermore, let $T'_-$ (resp. $T'_+$) be
+the tree obtained from $F_-$ (resp. $F_+$) by throwing away the leaves and the open half edge
+connecting to the leaves. Then let $A_- = q^{-1}(T'_+)$ and $A_+ = q^{-1}(T'_+)$. We have $A_-$ and $A_+$ are
+both admissible subsurfaces of $\Sigma_{d,r}^\infty$. Now one can produce a homeomorphism $\varphi_0: A_- \to A_+$
+which is identity on the based boundary and maps the admissible loops of $A_-$ to the admissible
+loops of $A_+$ following the information from $\rho$, mapping the visible part to the visible part for
+each admissible loop. From here, we extend $\varphi_0$ to a map $\varphi: \Sigma_{d,r}^\infty \to \Sigma_{d,r}^\infty$ such that $\varphi$ is a
+asymptotically rigid homeomorphism.
+
+If an element $g \in BV_{d,r}(\Sigma)$ is mapped to a trivial element $\pi(g) = [(F_-, \rho, F_+)] \in V_{d,r}$, then
+the two forests $F_-$ and $F_+$ are the same and the induced map $\rho$ is trivial. This means we can
+assume the support $A$ for the asymptotically rigid homeomorphism $\varphi_g$ corresponding to $g$ is
+the same as $\varphi(A)$ and $\varphi$ induces identity map on the admissible boundary components. Thus
+$g \in PMap_c(\Sigma_{d,r}^\infty)$. Finally, given any element $g \in PMap_c(\Sigma_{d,r}^\infty)$, it is clear that $\pi \circ j(g) = 1$. $\square$
+
+The mapping class group $\mathrm{Map}(\Sigma_{d,r}^{\infty})$ has a natural quotient topology coming from the compact-open topology on $\mathrm{Homeo}^{+}(\Sigma_{d,r}^{\infty}, \partial\Sigma_{d,r}^{\infty})$. See [AV20, Section 2.3, 4.1] for more information. In [AF17, Theorem 1.3], Aramayona and Funar showed that when $\Sigma$ is a closed surface, $\mathcal{H}\mathrm{V}_{2,1}(\Sigma)$ is dense in $\mathrm{Map}(\Sigma_{2,1}^{\infty})$. We improve their result to the following.
+
+**Theorem 3.19.** *The groups* $\mathcal{B}V_{d,r}(\Sigma)$ *and* $\mathcal{H}V_{d,r}(\Sigma)$ *are dense in the mapping class group*
+$\mathrm{Map}(\Sigma_{d,r}^{\infty})$.
+
+**Proof.** The proof in [AF17, Section 6] adapts directly to show that $\mathcal{H}V_{d,r}(\Sigma)$ is dense in $\mathrm{Map}(\Sigma_{d,r}^{\infty})$ and so we will not repeat it here. To show $\mathcal{B}V_{d,r}(\Sigma)$ is also dense in $\mathrm{Map}(\Sigma_{d,r}^{\infty})$, it suffices to show any element in $\mathcal{H}V_{d,r}(\Sigma)$ can be approximated by a sequence of elements in $\mathcal{B}V_{d,r}(\Sigma)$. Since $\mathcal{H}V_{d,r}(\Sigma)$ can be generated by $\mathcal{B}V_{d,r}(\Sigma)$ and half Dehn twists around the admissible loops in $\Sigma_{d,r}^{\infty}$, it suffices to show that any half Dehn twists around an admissible loop in $\Sigma_{d,r}^{\infty}$ can be approximated by a sequence of elements in $\mathcal{B}V_{d,r}(\Sigma)$. Given a admissible loop $L$, let $h_L$ be a half Dehn twist at $L$. We will construct a sequence of elements $x_i \in \mathcal{B}V_{d,r}(\Sigma)$ such that for any compact subset $K$ of $\Sigma_{d,r}^{\infty}$, there exists $N$ such that for any $j \ge N$, $x_j$ and $h_L$ coincide on $K$. Recall we have the map $q: \Sigma_{d,r}^{\infty} \to \mathcal{T}_{d,r}$ (cf. Definition 3.16) such that the admissible loops are mapped to edge middle points in $\mathcal{T}_{d,r}$. Now consider those admissible loops such that their image under $q$ lying below $q(L)$ have distance $i$ to $q(L)$. Note that there are $d^i$ such admissible loops. We list them as $L_{i,1}, \cdots, L_{i,d^i}$. Let $h_{L_{i,k}}$ be the half Dehn twists around $L_{i,k}$ and let $x_i = h_L h_{L_{i,1}} \cdots h_{L_{i,d^i}}$, then $x_i \in \mathcal{B}V_{d,r}(\Sigma)$ and the sequence $\{x_i\}$ has the desired property. $\square$
+
+Now recall by Remark 3.5 that $\Sigma_{d,r}^\infty$ is homeomorphic to $\Sigma \setminus C$ for any $d$ and $r$, hence we have
+the following corollary.
+
+**Corollary 3.20.** Let $\Sigma$ be any compact surface and $C$ be a Cantor set which lies in the interior of a disk in $\Sigma$. Then the mapping class group $\mathrm{Map}(\Sigma \setminus C)$ contains the following two families of dense subgroups: the asymptotic mapping class groups $\mathcal{B}V_{d,r}(\Sigma)$ which surject to the Higman-Thompson group $V_{d,r}$, and the half-twist asymptotic mapping class groups $\mathcal{H}V_{d,r}(\Sigma)$ which surject to the symmetric Higman-Thompson group $V_{d,r}(Z/2Z).
+
+**3.3. The asymptotic mapping class group of the disk punctured by the Cantor set.**
+
+In the last subsection, we want to identify the asymptotic mapping class group $\mathcal{B}V_{d,r}(D)$ with
+the oriented ribbon Higman–Thompson groups $\mathcal{R}\mathcal{V}_{d,r}^{+}$ and the half-twist asymptotic mapping
+class group $\mathcal{H}\mathcal{V}_{d,r}(D)$ with the ribbon Higman–Thompson group $\mathcal{R}\mathcal{V}_{d,r}$. The following lemma
+appears in [BT12, Section 2] without a proof, so we provide the details here.
+
+**Lemma 3.21.** Let $D_k$ be the $(k+1)$-holed sphere. Then $\mathrm{Map}(D_k)$ can be naturally identified with the pure oriented ribbon braid group $\mathrm{PRB}_k^+$.
+
+**Proof.** Note that $D_k$ can be identified with a disk with $k$ holes. Let $\partial_b$ denote the boundary of the disk. Let $\bar{D}_k$ be a disk with $k$ punctures obtained from $D_k$ by attaching one punctured disk
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+to each hole. The induced map $Cap : Map(D_k) \to PMap(\bar{D}_k)$ is the capping homomorphism. Note that $PMap(\bar{D}_k) \cong PB_k$. Now applying [FM11, Proposition 3.19] and the fact that the Dehn twists around the holes of $D_k$ commute, one sees that the kernel $K$ is a free abelian group of rank $k$ generated by these $k$ Dehn twists. Here the capping homomorphism splits. To prove this, we first embed $PB_k$ into $PRB_k^+$ by viewing the pure braid group of $k$-strings as the set of ribbon braids on $k$ bands such that the bands have no twists. We can think of $D_k$ as being embedded into $\mathbb{R}^2$ with $\partial_b$ as the unit circle and the $k$ holes in $D_k$ equally distributed inside $\partial_b$ along the $x$-axis. The intersections of these holes with the $x$-axis gives $k$ sub-intervals of the $x$-axis denoted $I_1, \cdots, I_k$. We now put the bands representing a pure braid $x \in PB_k \le PRB_k^+$ in $D \times [0, 1]$ which starts and ends at $I_1, \cdots, I_k$. Note that the bands here will not twist at all. Now we comb the bands straight from bottom to top. This induces a homeomorphism of $D_k \times \{0\}$ and hence an element in the mapping class group $Map(D_k)$. One checks that this map is a group homomorphism and injective. Since $PB_k$ acts on $K$ trivially, we have $Map(D_k) \cong K \times PB_k \cong \mathbb{Z}^k \times PB_k \cong PRB_k^+$ where the number of Dehn twists around each boundary component is naturally identified with the number of full twists on each bands. $\square$
+
+To promote Lemma 3.21 such that it works for the ribbon braid group, we need some extra terminology. As in the proof of Lemma 3.21, we identify $D_k$ with the unit disk in $\mathbb{R}^2$ with $k$ small disks whose centers are equally distributed on the $x$-axis removed. The $x$-axis cuts the boundary loops of each deleted disk into two components, providing a cell structure on the loops. We will call the part that lies above the $x$-axis the *visible part*. We define the *rigid mapping class group* RMap$_+(D_k)$ of $D_k$ to be the isotopy classes of homeomorphisms of $D_k$ which fix $\partial_b D_k$ pointwise and map the visible part of the holes to the visible part of the holes. Note elements in $\partial_b D_k$ are allowed to map one boundary hole to another just as in the definition of the asymptotic mapping class group. If we only assume the cell structure on the loops has to be preserved, the resulting group is called *quasi-rigid mapping class group* $D_k$ and denoted by $RMap(D_k)$. With these preparations, the following lemma is now clear.
+
+**Lemma 3.22.** There is a natural isomorphism between the oriented ribbon braid group $RB_k^+$ and $RMap_+(D_k)$ (resp. between the ribbon braid group $RB_k$ and $RMap(D_k)$).
+
+**Proof.** As in the proof of Lemma 3.21, we put the element in the (oriented) ribbon braid group between $D \times [0, 1]$, then we comb the bands straight from bottom to top which gives the corresponding element in $RMap_+(D_k)$ (resp. $RMap(D_k)$). $\square$
+
+Given two admissible subsurfaces $A$ and $A'$ of $D_{d,r}^\infty$ (possibly with different $r$) with $k$ admissible boundary components, we want to fix a canonical way to identify a homeomorphism $f: A \to A'$ as an element in the ribbon braid group. Note that each boundary loop except the base one inherits a visible side from $D_{d,r}^\infty$. We will use the puncture model for $D_{d,r}^\infty$ going forward.
+
+As above, let $D_k$ be the subsurface of $D$ which is the complement of $k$ disjoint open disks with centers at $a_i = \frac{2i-k-1}{k+1}$ of radius $2^{-k}$ for $1 \le i \le k$. Now given any admissible subsurface $A_k$ of $D_{d,r}^\infty$ with $k$ many admissible boundaries, denote the centers from left to right by $c_i \in [0, 1] \times \{0\}$, $1 \le i \le k$ with radius $r_1, r_2, \dots, r_k$. Now we define an isotopy $\mathcal{N}_{A_k}: D \times [0, 1] \to D$ such that $\mathcal{N}_{A_{k,0}} = \text{id}_D$ and $\mathcal{N}_{A_{k,1}}$ maps $A_k$ to $D_k$ via a homeomorphism. We first shrink the admissible boundary loops of $A_k$ such that they have radius $r$, where $r = \min\{r_1, \dots, r_k, 2^{-k}\}$. Then we isotope $A_k$ by moving the centers $c_i$ to $a_i$ along $[0, 1] \times \{0\}$ in $D$. And in the last step we enlarge the radius one by one to $2^{-k}$. The following lemma is now immediate.
+
+**Lemma 3.23.** Let $\phi : D_{d,r}^\infty \to D_{d,r}^\infty$ be an asymptotically rigid (resp. quasi-rigid) homeomorphism which is supported on the admissible subsurface $A_k$. Denote $\phi'_k = \phi(A_k)$, then
+
+(1) $\mathbf{r}_\phi = \mathcal{N}_{A'_{k,1}} \circ \phi|_{A_k} \circ \mathcal{N}_{A_{k,1}}^{-1} : D_k \to D_k$ gives an element in the oriented ribbon braid group $RB_k^+$ (resp. the ribbon braid group $RB_k$). Conversely, given an element $\mathbf{r} \in RB_k^+$ (resp. $RB_k$), we have an asymptotic rigid (resp. quasi-rigid) homeomorphism which is unique up to isotopy, supported on $A_k$, and map $A_k$ to $\phi'_k$.
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+**Definition 4.2** ([RWW17, Definition 1.5]). Let $(\mathcal{C}, \oplus, 0)$ be a monoidal category with 0 initial. We say that $\mathcal{C}$ is *prebraided* if its underlying groupoid is braided and for each pair of objects $A$ and $B$ in $\mathcal{C}$, the groupoid braiding $b_{A,B}: A \oplus B \to B \oplus A$ satisfies
+
+$$ b_{A,B} \circ (A \oplus \iota_B) = \iota_B \oplus A : A \to B \oplus A. $$
+
+**Definition 4.3.** [RWW17, Definition 2.1] Let $(\mathcal{C}, \oplus, 0)$ be a monoidal category with 0 initial and $(A, X)$ a pair of objects in $\mathcal{C}$. Define $W_n(A, X)_\bullet$ to be the semi-simplicial set with set of $p$-simplices
+
+$$ W_n(A, X)_p := \operatorname{Hom}_\mathcal{C}(X^{\oplus p+1}, A \oplus X^{\oplus n}) $$
+
+and with face map
+
+$$ d_i : \operatorname{Hom}_{\mathcal{C}}(X^{\oplus p+1}, A \oplus X^{\oplus n}) \to \operatorname{Hom}_{\mathcal{C}}(X^{\oplus p}, A \oplus^{\oplus n}) $$
+
+defined by precomposing with $X^{\oplus i} \oplus \iota_X \oplus X^{\oplus p-i}$.
+
+Also define the following property for a fixed pair $(A, X)$ and a slope $k \ge 2$.
+
+**LH3** For all $n \ge 1$, $W_n(A, X)_\bullet$ is $(\frac{n-2}{k})$-connected.
+
+Quite often, we can reduce the semi-simplicial complex to a simplicial complex.
+
+**Definition 4.4** ([RWW17, Definition 2.8]). Let $A, X$ be objects of a homogeneous category $(\mathcal{C}, \oplus, 0)$. For $n \ge 1$, let $S_n(A, X)$ denote the simplicial complex whose vertices are the maps $f: X \to A \oplus X^{\oplus n}$ and whose $p$-simplices are $(p+1)$-sets $\{f_0, \dots, f_p\}$ such that there exists a morphism $f: X^{\oplus p+1} \to A \oplus X^{\oplus n}$ with $f \circ i_j = f_j$ for some order on the set, where
+
+$$ i_j = \iota_{X \oplus j} \oplus \mathrm{id}_X \oplus \iota_{X^{\oplus p-j}} : X = 0 \oplus X \oplus 0 \to X^{\oplus p+1}. $$
+
+**Definition 4.5.** Let $\mathrm{Aut}(A \oplus X^{\oplus\infty})$ be the colimit of
+
+$$ \dots \xrightarrow{-\oplus X} \mathrm{Aut}(A \oplus X^{\oplus n}) \xrightarrow{-\oplus X} \mathrm{Aut}(A \oplus X^{\oplus n+1}) \xrightarrow{-\oplus X} \mathrm{Aut}(A \oplus X^{\oplus n+2}) \xrightarrow{-\oplus X} \dots $$
+
+Then any $\mathrm{Aut}(A \oplus X^{\oplus\infty})$-module $M$ may be considered as an $\mathrm{Aut}(A \oplus X^{\oplus n})$-module for any $n$, by restriction, which we continue to call $M$. We say that the module $M$ is abelian if the action of $\mathrm{Aut}(A \oplus X^{\oplus\infty})$ on $M$ factors through the abelianizations of $\mathrm{Aut}(A \oplus X^{\oplus\infty})$, or in other words if the derived subgroup of $\mathrm{Aut}(A \oplus X^{\oplus\infty})$ acts trivially on $M$.
+
+We are now ready to quote the theorem that we will use.
+
+**Theorem 4.6** ([RWW17, Theorem 3.1]). Let $(\mathcal{C}, \oplus, 0)$ be a pre-braided homogeneous category satisfying **LH3** for a pair $(A, X)$ with slope $k \ge 3$. Then for any abelian $\mathrm{Aut}(A \oplus X^{\oplus\infty})$-module $M$ the map
+
+$$ H_i(\mathrm{Aut}(A \oplus X^{\oplus n}); M) \to H_i(\mathrm{Aut}(A \oplus X^{\oplus n+1}); M) $$
+
+induced by the natural inclusion map is surjective if $i \le \frac{n-k+2}{k}$, and injective if $i \le \frac{n-k}{k}$.
+
+**4.2. Homogeneous category for the groups** $RV_{d,r}^+$. The purpose of this section is to produce a homogeneous category for proving homological stability of the ribbon Higman–Thompson groups $RV_{d,r}^+$. Note that by Theorem 3.24, it is same as proving the asymptotic mapping class groups $BV_{d,r}$ have homological stability. This allows us to define our homogeneous category geometrically. The category is similar to the ones produced in [RWW17, Section 5.6]. Essentially, we replace the annulus or Möbius band by the infinite surface $D_{d,1}^\infty$.
+
+Recall $D_{d,r}^\infty$ is an infinite surface equipped with a canonical asymptotic rigid structure. Let $I = [-1, 1] \subset \partial_b D_{d,r}^\infty$ be an embedded interval. Let $I^- = [-1, 0]$ and $I^+ = [0, 1]$ be subintervals of $I$. Let $D_{d,1}^\infty \oplus D_{d,1}^\infty$ be the boundary sum of two copies of $D_{d,1}^\infty$ obtained by identifying $I^+$ of the first copy with $I^-$ of the second copy. Inductively, we could define similarly $\oplus_r D_{d,1}^\infty$ for any $r \ge 0$. Here $\oplus_0 D_{d,1}^\infty$ is just the standard disk $D$. Abusing notation, when referring to $I^-$ and $I^+$ on $\oplus_r D_{d,1}^\infty$, we will mean the two copies of $I^-$ and $I^+$ which remain on the boundary. Thus we have an operation $\oplus$ on the set $\oplus_r D_{d,1}^\infty$ for any $r \ge 0$. See Figure 6(b) for a picture of $(\oplus_2 D_{d,1}^\infty) \oplus (\oplus_3 D_{d,1}^\infty)$. In fact, we have (($\oplus_r D_{d,1}^\infty$), $\oplus$) is the free monoid generated by $D_{d,1}^\infty$. Note
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+FIGURE 6. The braided monoidal structure for the category $\mathcal{G}_d$
+
+that $\oplus_r D_{d,1}^\infty$ has a naturally induced $d$-rigid structure and we can identify it with $D_{d,r}^\infty$, which will be of use to us later.
+
+We can now define the category $\mathcal{G}_d$ to be the monoidal category with objects $\oplus_r D_{d,1}^\infty$, $r \ge 0$, $\oplus$ as the operation, and $D$ as the $0$ object. So far it is the same as defining the objects as the natural numbers and addition as the operation. When $r=s$, we define the morphisms $\text{Hom}(\oplus_r D_{d,1}^\infty, \oplus_s D_{d,1}^\infty) = \mathcal{B}\text{V}_{d,r}$ which is the group of isotopy classes of asymptotically rigid homeomorphisms of $D_{d,r}^\infty$; when $r \ne s$, let $\text{Hom}(\oplus_r D_{d,1}^\infty, \oplus_s D_{d,1}^\infty) = \emptyset$. Note that we did not universally define the morphisms to be the sets of isotopy classes of asymptotically rigid homeomorphisms as we want our category to satisfy cancellation, i.e., if $A \oplus C = A$ then $C = 0$, see [RWW17, Remark 1.11] for more information. The category $\mathcal{G}_d$ has a natural braiding as in the usual braid group case, see Figure 6(c).
+
+Now applying [RWW17, Theorem 1.10], we have a homogeneous category $\mathcal{U}\mathcal{G}_d$. The category $\mathcal{U}\mathcal{G}_d$ has the same objects as $\mathcal{G}_d$ and morphisms defined as following: For any $s \le r$, a morphism in $\text{Hom}(\oplus_s D_{d,1}^\infty, \oplus_r D_{d,1}^\infty)$ is an equivalence class of pairs $(\oplus_{r-s} D_{d,1}^\infty, f)$ where $f: (\oplus_{r-s} D_{d,1}^\infty) \oplus (\oplus_s D_{d,1}^\infty) \to \oplus_r D_{d,1}^\infty$ is a morphism in $\mathcal{G}_d$ and $(\oplus_{r-s} D_{d,1}^\infty, f) \sim (\oplus_{r-s} D_{d,1}^\infty, f')$ if there exists an isomorphism $g: \oplus_{r-s} D_{d,1}^\infty \to \oplus_{r-s} D_{d,1}^\infty \in \mathcal{G}_d$ making the diagram commute up to isotopy.
+
+We write $[\oplus_{s-r} D_{d,1}^{\infty}, f]$ for such an equivalence class. Now by Theorem 4.6, to prove the homological stability for the oriented ribbon Higman–Thompson groups, we only need to verify Condition **LH3**, i.e. the complex $W_r(D_{d,1}^{\infty}, D_{d,1}^{\infty})$ is highly connected. As a matter of fact, we will show that $W_r(D, D_{d,1}^{\infty})\cdot$ is $(r-3)$-connected in the next subsection. First, let us further characterize the morphisms in $\mathcal{U}\mathcal{G}_d$. Call $0 = I^- \cap I^+$ the basepoint of $\oplus_r D_{d,1}^\infty$.
+
+**Definition 4.7.** Given $s < r$, an injective map $\varphi : (\oplus_s D_{d,1}^\infty, I^+) \to (\oplus_r D_{d,1}^\infty, I^+)$ is called an *asymptotically rigid embedding* if it satisfies the following properties:
+
+(1) $\varphi(\partial D_{d,s}^\infty) \cap \partial D_{d,r}^\infty = I^+$.
+
+(2) $\varphi$ maps $\oplus_s D_{d,1}^\infty$ homeomorphically to $\varphi(\oplus_s D_{d,1}^\infty)$ and there exists an admissible surface $A \subset \oplus_s D_{d,1}^\infty$ such that $\varphi: \oplus_s D_{d,1}^\infty \setminus A \to \varphi(\oplus_s D_{d,1}^\infty) \setminus \varphi(A)$ is rigid.
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+(3) The closure of the complement of $\varphi(\oplus_s D_{d,1}^\infty)$ in $\oplus_r D_{d,1}^\infty$ with its induced $d$-rigid structure is asymptotically rigidly homeomorphic to $\oplus_{r-s} D_{d,1}^\infty$.
+
+**Lemma 4.8.** For $s < r$, the equivalence classes of pairs $[\oplus_{r-s} D_{d,1}^\infty, f]$ are in one-to-one correspondence with the isotopy classes of asymptotically rigid embeddings of $(\oplus_s D_{d,1}^\infty, I^+)$ into $(\oplus_r D_{d,1}^\infty, I^+)$.
+
+**Remark 4.9.** Here isotopies are carried out among asymptotically rigid embeddings.
+
+*Proof.* Given an equivalence class of a pair $[\oplus_{t-s}D_{d,1}^{\infty}, f]$, we have the restriction map $f|_{\oplus_s D_{d,1}^{\infty}}$ is an asymptotically rigid embedding. Any two equivalence classes of pairs will induce the same map $f|_{\oplus_s D_{d,1}^{\infty}}$, hence we have a well-defined map from the set of equivalence pairs to the set of isotopy classes of asymptotically rigid embeddings.
+
+We produce the inverse of the restriction map as follows. If we have an asymptotically rigid embedding $\varphi : (\oplus_s D_{d,1}^\infty, I^+) \to (\oplus_r D_{d,1}^\infty, I^+)$, by part 3 of Definition 4.7, we also have an asymptotically rigid homeomorphism $\phi : C \to \oplus_{r-s} D_{d,1}^\infty$ where $C$ is the closure of the complement of $\varphi(\oplus_s D_{d,1}^\infty)$ in $\oplus_r D_{d,1}^\infty$. Up to isotopy, we can assume $\phi^{-1}|_{I^+}$ coincides with $\varphi|_{I^-}$. Now define a map $\tilde{f} : (\oplus_{r-s} D_{d,1}^\infty) \oplus (\oplus_s D_{d,1}^\infty) \to \oplus_r D_{d,1}^\infty$ by $\tilde{f}|_{\oplus_{r-s} D_{d,1}^\infty} = \phi^{-1}$ and $\tilde{f}|_{\oplus_s D_{d,1}^\infty} = \varphi$. One can check that $\tilde{f}$ is an asymptotically rigid homeomorphism. Then $(\oplus_{r-s} D_{d,1}^\infty, \tilde{f})$ gives a representative of an equivalence class of pairs. $\square$
+
+4.3. **Higher connectivity of the complex W