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.