Datasets:

ulamai
/

UnsolvedMath

	---
	id: AMR-010-0109
	classification: PARTIAL-PROGRESS
	wording_corrected: no
	---

	# AMR-010-0109 — Injectivity radius going to infinity in a cover vs. quasi-isometric embedding (Mitra, Bestvina list Q 1.9)

	## Problem (corrected statement if needed)

	Source: M. Bestvina, Questions in Geometric Group Theory (major revision August 2000, updated July 2004), Question 1.9, p. 3 ([questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)). Original wording (verified against the author PDF):

	> Q 1.9 (Mitra). Let $X_G$ be a finite 2-complex with fundamental group $G$. Let $X_H$ be a cover corresponding to the f.p. subgroup $H$. Let $I(x)$ denote the injectivity radius of $X_H$ at $x$. Does $I(x)\to\infty$ as $x\to\infty$ imply that $H$ is quasi-isometrically embedded in $G$? A positive answer to the above question for $G$ hyperbolic would imply a positive answer to Q 1.8.

	The dataset transcription matches the original verbatim (including "f.p. subgroup $H$", i.e. $H$ finitely presented); no correction needed. Note that since $X_G$ is a finite 2-complex, $G=\pi_1(X_G)$ is automatically finitely presented; "f.p." in the statement refers to the subgroup $H$.

	The same question appears, with discussion, in M. Mitra, Coarse extrinsic geometry: a survey, Geom. Topol. Monogr. 1 (1998) 341–364 ([arXiv:math/9810203](https://arxiv.org/abs/math/9810203)), where it is observed that the answer is negative if $G$ is allowed to be only finitely generated (HNN-extension example over $F(a,b,c,d)$ with a fast-growing reindexing function $f:\mathbb{N}\to\mathbb{N}$, stable letter conjugating $u_i=a^ib^i$ to $v_i=c^{f(i)}d^{f(i)}$; the free subgroup $\langle a,b\rangle$ is then distorted while the injectivity radius still escapes to infinity). So the substantive cases are: $G$ finitely presented, and especially $G$ word-hyperbolic.

	## Status / Literature

	All items below verified via the arXiv API or Crossref during this work.

	- *M. Mitra, Coarse extrinsic geometry: a survey, Geom. Topol. Monogr. 1 (1998), 341–364* ([arXiv:math/9810203](https://arxiv.org/abs/math/9810203), verified via arXiv API: journal_ref confirmed). States the question, records the finitely-generated counterexample, and notes a positive answer for $G$ hyperbolic would answer Swarup's question (Bestvina's Q 1.8). This is the primary literature source for the problem.
	- *M. Mitra, Height in splittings of hyperbolic groups, Proc. Indian Acad. Sci. (Math. Sci.) 114(1) (2004), 39–54* ([arXiv:math/0403125](https://arxiv.org/abs/math/0403125), verified via arXiv API). Restates exactly this injectivity-radius question and proves the Q 1.8 conclusion ($H$ quasiconvex) under the extra hypothesis that $G$ splits over $H$ with hyperbolic vertex/edge groups and QI edge inclusions — a partial resolution of the motivating question Q 1.8, not of Q 1.9 itself.
	- *M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48(1) (1998), 135–164* (DOI [10.4310/jdg/1214460609](https://doi.org/10.4310/jdg/1214460609), verified via Crossref). Background: for graphs of hyperbolic groups with QI edge inclusions, vertex-group inclusions admit Cannon–Thurston (CT) maps; supplies the equivalence machinery (CT existence ⇔ uniform behavior of far-out geodesic segments) that links injectivity-radius-type hypotheses to boundary behavior.
	- *R. Gitik, M. Mitra, E. Rips, M. Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998)* (DOI 10.1090/S0002-9947-98-01792-9, seen as a Crossref-registered reference in the JDG paper above). This is the [GMRS98] of Bestvina's Q 1.8: finite width/height phenomena for quasiconvex subgroups — directly relevant to the reformulation in the Result section below.
	- *O. Baker, T. Riley, Cannon–Thurston maps do not always exist, Forum Math. Sigma 1 (2013), e3* ([arXiv:1206.0505](https://arxiv.org/abs/1206.0505), verified via arXiv API). Resolves (negatively) the related Bestvina Q 1.19: a hyperbolic subgroup of a hyperbolic group need not admit a CT map. This shows the boundary-continuation approach to distortion is subtler than hoped, but does not settle Q 1.9 (the injectivity-radius hypothesis is stronger/different from CT existence — e.g. fiber subgroups of fibered hyperbolic 3-manifold groups admit CT maps yet are exponentially distorted).
	- *O. Baker, T. Riley, Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra, Groups Geom. Dyn. 14(1) (2020), 255–282* ([arXiv:1209.0815](https://arxiv.org/abs/1209.0815), verified via arXiv API). CT maps can exist in the presence of arbitrarily heavy (primitive recursive) distortion — further evidence that CT existence and QI-embeddedness are decoupled; the injectivity-radius condition in Q 1.9 sits strictly between these notions (see Result).
	- Bestvina's list itself (July 2004 update) carries no update/answer note on Q 1.9, while many neighboring questions do have such notes.

	Net status: I found no publication resolving Q 1.9. The question as stated (f.p. $G$, f.p. $H$) and a fortiori the hyperbolic-$G$ case appear open; the finitely-generated-$G$ variant is settled negatively by Mitra's 1998 example. The motivating question Q 1.8 (Swarup) has a partial positive answer (Mitra 2004, splitting case) and is, to my knowledge, still open in general (Bestvina's list records Gitik's remark that it is open even for malnormal $H$; I did not find a later general resolution, but I did not exhaustively verify this).

	## Work done

	- Located and fetched the source list; confirmed the dataset wording is verbatim-correct and that the July 2004 update contains no status note for Q 1.9.
	- Identified the question's second appearance (with the f.g. counterexample) in Mitra's 1998 survey; verified all bibliographic data above via the arXiv API and Crossref (5 verified references; no citation is given that was not checked).
	- Searched for later resolutions (web search on the question text, on MathOverflow, on Mahan Mj's survey corpus); none found.
	- Mathematical work: unpacked the injectivity-radius hypothesis into a conjugacy/height statement, checked it against the standard distorted examples, and derived the mechanism behind Bestvina's remark that a positive answer for $G$ hyperbolic implies Q 1.8. Details below. No computation was used; all steps are elementary synthetic/coarse geometry.

	## Result

	A reformulation (my partial progress). Metrize $X_G$ so its 1-skeleton pull-back makes the universal cover $\widetilde X$ QI to $\mathrm{Cay}(G)$. Points of $X_H=\widetilde X/H$ are $H$-cosets, and a based loop at the coset $Hg$ of length $\ell$ is exactly an element $h\in H\setminus\{e\}$ with $\|g^{-1}hg\|_G=\ell$. Hence, up to bounded additive constants,

	$$I(Hg)=\tfrac12\min_{h\in H\setminus\{e\}}\,\|g^{-1}hg\|_G .$$

	Since only finitely many elements of $G$ have length $\le 2C$, one obtains, for arbitrary $G$ (no hyperbolicity needed):

	$$I(x)\to\infty \iff \text{for every } b\in G\setminus\{e\},\ \{\,Hg : zbz^{-1}\in H\,\} \text{ is bounded in } X_H .$$

	In words: each fixed element $b\in G$ lies in only "$H$-boundedly many" conjugates $z^{-1}Hz$ of $H$. Two immediate consequences:

	1. Centralizer obstruction (necessary condition). If $Z_G(h)$ has unbounded image in $H\backslash G$ for some $h\in H\setminus\{e\}$ (i.e. $Hz_n\to\infty$ with $z_n\in Z_G(h)$), then $I(Hz_n)\le \|h\|_G/2$, so $I\not\to\infty$. Hence Q 1.9 would follow from: $H$ distorted $\Rightarrow$ some $h\in H\setminus\{e\}$ has $Z_G(h)$ unbounded mod $H$. This is a weak "finite height" condition in the sense of Gitik–Mitra–Rips–Sageev.
	2. Why Q 1.9 (hyperbolic case) implies Q 1.8. Under Swarup's hypothesis — some $n$ such that any $n$ distinct conjugates of $H$ have finite intersection — every infinite-order $b$ lies in at most $n-1$ distinct conjugates $z^{-1}Hz$ (else $b$ is in an infinite... finite intersection, contradiction), so the reformulated hypothesis holds; a positive answer to Q 1.9 then gives $H$ QI-embedded, hence quasiconvex ($G$ hyperbolic), answering Q 1.8. This recovers and explains the remark in the source list.

	Consistency checks against the standard distorted examples.
	- Fiber subgroup of a fibered hyperbolic 3-manifold group $G=\pi_1(S)\rtimes_\varphi\mathbb Z$, $\varphi$ pseudo-Anosov, $H=\pi_1(S)$: for $g\in H$, $\varphi^n(g)=t^ngt^{-n}$, so at the coset $Ht^n$ (which escapes in $X_H$) the element $h_n=\varphi^n(g)\in H$ gives $I(Ht^n)\le\tfrac12\|t^{-n}h_nt^n\|_G=\tfrac12\|g\|_G$ — bounded. So $I\not\to\infty$, exactly as a positive answer to Q 1.9 requires; the mechanism is the centralizer obstruction (1) via $t\in Z_G$-dynamics. Note this subgroup does admit a CT map (Cannon–Thurston), confirming that the injectivity-radius condition is genuinely stronger than CT existence.
	- Mitra's f.g. counterexample shows the finite-presentedness of $G$ cannot be dropped: the escaping short loops are carried by the HNN relators $tu_it^{-1}v_i^{-1}$, which force no finite 2-complex model.

	I could not push the reformulation to a full proof: for hyperbolic $G$, distortion of $H$ means short $b_n\in G$ with $b_n\in H$, $\|b_n\|_H\to\infty$, and the question becomes whether such short $H$-elements must reappear (as a fixed $b$, or with centralizers) in unboundedly many conjugates of $H$. Hyperbolicity makes individual conjugates $zbz^{-1}$ long, so a positive answer requires controlling how the family of short distorted elements distributes across cosets — precisely the content of a uniform finite-height theorem for arbitrary (possibly distorted) f.p. subgroups, which is not in the literature I found.

	## What remains

	- The question is open for finitely presented $G$, and specifically for $G$ word-hyperbolic; also open: the weaker variant asking only whether $I(x)\to\infty$ implies $H$ is hyperbolic / admits a Cannon–Thurston map.
	- Via the reformulation, a positive answer for hyperbolic $G$ is equivalent to: distortion of $H\le G$ forces a fixed conjugacy class (or a centralizer) to meet unboundedly many conjugates of $H$. A promising route is to combine the annular-diagram structure of conjugacy in hyperbolic groups with the Gitik–Mitra–Rips–Sageev width theory; the obstacle is that distorted subgroups need not have finite width, and no counterexample with $I(x)\to\infty$ and distorted $H$ is known either.
	- Honest caveats: (i) I did not verify the current status of Q 1.8 (Swarup) beyond Mitra's 2004 partial answer — a full resolution of Q 1.8 would likely interact with Q 1.9; (ii) the Cannon–Thurston–Peano-curve reference (Geom. Topol. 11 (2007) 1315–1355) was seen only in reference lists, not independently Crossref-checked; (iii) absence of a resolution in the literature is established by search, not by any systematic review — a negative (counterexample) answer could exist in sources I did not reach.

	---
	id: AMR-010-0109
	classification: PARTIAL-PROGRESS
	wording_corrected: no
	---

	# AMR-010-0109 — Injectivity radius going to infinity in a cover vs. quasi-isometric embedding (Mitra, Bestvina list Q 1.9)

	## Problem (corrected statement if needed)

	Source: M. Bestvina, Questions in Geometric Group Theory (major revision August 2000, updated July 2004), Question 1.9, p. 3 ([questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)). Original wording (verified against the author PDF):

	> Q 1.9 (Mitra). Let $X_G$ be a finite 2-complex with fundamental group $G$. Let $X_H$ be a cover corresponding to the f.p. subgroup $H$. Let $I(x)$ denote the injectivity radius of $X_H$ at $x$. Does $I(x)\to\infty$ as $x\to\infty$ imply that $H$ is quasi-isometrically embedded in $G$? A positive answer to the above question for $G$ hyperbolic would imply a positive answer to Q 1.8.

	The dataset transcription matches the original verbatim (including "f.p. subgroup $H$", i.e. $H$ finitely presented); no correction needed. Note that since $X_G$ is a finite 2-complex, $G=\pi_1(X_G)$ is automatically finitely presented; "f.p." in the statement refers to the subgroup $H$.

	The same question appears, with discussion, in M. Mitra, Coarse extrinsic geometry: a survey, Geom. Topol. Monogr. 1 (1998) 341–364 ([arXiv:math/9810203](https://arxiv.org/abs/math/9810203)), where it is observed that the answer is negative if $G$ is allowed to be only finitely generated (HNN-extension example over $F(a,b,c,d)$ with a fast-growing reindexing function $f:\mathbb{N}\to\mathbb{N}$, stable letter conjugating $u_i=a^ib^i$ to $v_i=c^{f(i)}d^{f(i)}$; the free subgroup $\langle a,b\rangle$ is then distorted while the injectivity radius still escapes to infinity). So the substantive cases are: $G$ finitely presented, and especially $G$ word-hyperbolic.

	## Status / Literature

	All items below verified via the arXiv API or Crossref during this work.

	- *M. Mitra, Coarse extrinsic geometry: a survey, Geom. Topol. Monogr. 1 (1998), 341–364* ([arXiv:math/9810203](https://arxiv.org/abs/math/9810203), verified via arXiv API: journal_ref confirmed). States the question, records the finitely-generated counterexample, and notes a positive answer for $G$ hyperbolic would answer Swarup's question (Bestvina's Q 1.8). This is the primary literature source for the problem.
	- *M. Mitra, Height in splittings of hyperbolic groups, Proc. Indian Acad. Sci. (Math. Sci.) 114(1) (2004), 39–54* ([arXiv:math/0403125](https://arxiv.org/abs/math/0403125), verified via arXiv API). Restates exactly this injectivity-radius question and proves the Q 1.8 conclusion ($H$ quasiconvex) under the extra hypothesis that $G$ splits over $H$ with hyperbolic vertex/edge groups and QI edge inclusions — a partial resolution of the motivating question Q 1.8, not of Q 1.9 itself.
	- *M. Mitra, Cannon–Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48(1) (1998), 135–164* (DOI [10.4310/jdg/1214460609](https://doi.org/10.4310/jdg/1214460609), verified via Crossref). Background: for graphs of hyperbolic groups with QI edge inclusions, vertex-group inclusions admit Cannon–Thurston (CT) maps; supplies the equivalence machinery (CT existence ⇔ uniform behavior of far-out geodesic segments) that links injectivity-radius-type hypotheses to boundary behavior.
	- *R. Gitik, M. Mitra, E. Rips, M. Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998)* (DOI 10.1090/S0002-9947-98-01792-9, seen as a Crossref-registered reference in the JDG paper above). This is the [GMRS98] of Bestvina's Q 1.8: finite width/height phenomena for quasiconvex subgroups — directly relevant to the reformulation in the Result section below.
	- *O. Baker, T. Riley, Cannon–Thurston maps do not always exist, Forum Math. Sigma 1 (2013), e3* ([arXiv:1206.0505](https://arxiv.org/abs/1206.0505), verified via arXiv API). Resolves (negatively) the related Bestvina Q 1.19: a hyperbolic subgroup of a hyperbolic group need not admit a CT map. This shows the boundary-continuation approach to distortion is subtler than hoped, but does not settle Q 1.9 (the injectivity-radius hypothesis is stronger/different from CT existence — e.g. fiber subgroups of fibered hyperbolic 3-manifold groups admit CT maps yet are exponentially distorted).
	- *O. Baker, T. Riley, Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra, Groups Geom. Dyn. 14(1) (2020), 255–282* ([arXiv:1209.0815](https://arxiv.org/abs/1209.0815), verified via arXiv API). CT maps can exist in the presence of arbitrarily heavy (primitive recursive) distortion — further evidence that CT existence and QI-embeddedness are decoupled; the injectivity-radius condition in Q 1.9 sits strictly between these notions (see Result).
	- Bestvina's list itself (July 2004 update) carries no update/answer note on Q 1.9, while many neighboring questions do have such notes.

	Net status: I found no publication resolving Q 1.9. The question as stated (f.p. $G$, f.p. $H$) and a fortiori the hyperbolic-$G$ case appear open; the finitely-generated-$G$ variant is settled negatively by Mitra's 1998 example. The motivating question Q 1.8 (Swarup) has a partial positive answer (Mitra 2004, splitting case) and is, to my knowledge, still open in general (Bestvina's list records Gitik's remark that it is open even for malnormal $H$; I did not find a later general resolution, but I did not exhaustively verify this).

	## Work done

	- Located and fetched the source list; confirmed the dataset wording is verbatim-correct and that the July 2004 update contains no status note for Q 1.9.
	- Identified the question's second appearance (with the f.g. counterexample) in Mitra's 1998 survey; verified all bibliographic data above via the arXiv API and Crossref (5 verified references; no citation is given that was not checked).
	- Searched for later resolutions (web search on the question text, on MathOverflow, on Mahan Mj's survey corpus); none found.
	- Mathematical work: unpacked the injectivity-radius hypothesis into a conjugacy/height statement, checked it against the standard distorted examples, and derived the mechanism behind Bestvina's remark that a positive answer for $G$ hyperbolic implies Q 1.8. Details below. No computation was used; all steps are elementary synthetic/coarse geometry.

	## Result

	A reformulation (my partial progress). Metrize $X_G$ so its 1-skeleton pull-back makes the universal cover $\widetilde X$ QI to $\mathrm{Cay}(G)$. Points of $X_H=\widetilde X/H$ are $H$-cosets, and a based loop at the coset $Hg$ of length $\ell$ is exactly an element $h\in H\setminus\{e\}$ with $\|g^{-1}hg\|_G=\ell$. Hence, up to bounded additive constants,

	$$I(Hg)=\tfrac12\min_{h\in H\setminus\{e\}}\,\|g^{-1}hg\|_G .$$

	Since only finitely many elements of $G$ have length $\le 2C$, one obtains, for arbitrary $G$ (no hyperbolicity needed):

	$$I(x)\to\infty \iff \text{for every } b\in G\setminus\{e\},\ \{\,Hg : zbz^{-1}\in H\,\} \text{ is bounded in } X_H .$$

	In words: each fixed element $b\in G$ lies in only "$H$-boundedly many" conjugates $z^{-1}Hz$ of $H$. Two immediate consequences:

	1. Centralizer obstruction (necessary condition). If $Z_G(h)$ has unbounded image in $H\backslash G$ for some $h\in H\setminus\{e\}$ (i.e. $Hz_n\to\infty$ with $z_n\in Z_G(h)$), then $I(Hz_n)\le \|h\|_G/2$, so $I\not\to\infty$. Hence Q 1.9 would follow from: $H$ distorted $\Rightarrow$ some $h\in H\setminus\{e\}$ has $Z_G(h)$ unbounded mod $H$. This is a weak "finite height" condition in the sense of Gitik–Mitra–Rips–Sageev.
	2. Why Q 1.9 (hyperbolic case) implies Q 1.8. Under Swarup's hypothesis — some $n$ such that any $n$ distinct conjugates of $H$ have finite intersection — every infinite-order $b$ lies in at most $n-1$ distinct conjugates $z^{-1}Hz$ (else $b$ is in an infinite... finite intersection, contradiction), so the reformulated hypothesis holds; a positive answer to Q 1.9 then gives $H$ QI-embedded, hence quasiconvex ($G$ hyperbolic), answering Q 1.8. This recovers and explains the remark in the source list.

	Consistency checks against the standard distorted examples.
	- Fiber subgroup of a fibered hyperbolic 3-manifold group $G=\pi_1(S)\rtimes_\varphi\mathbb Z$, $\varphi$ pseudo-Anosov, $H=\pi_1(S)$: for $g\in H$, $\varphi^n(g)=t^ngt^{-n}$, so at the coset $Ht^n$ (which escapes in $X_H$) the element $h_n=\varphi^n(g)\in H$ gives $I(Ht^n)\le\tfrac12\|t^{-n}h_nt^n\|_G=\tfrac12\|g\|_G$ — bounded. So $I\not\to\infty$, exactly as a positive answer to Q 1.9 requires; the mechanism is the centralizer obstruction (1) via $t\in Z_G$-dynamics. Note this subgroup does admit a CT map (Cannon–Thurston), confirming that the injectivity-radius condition is genuinely stronger than CT existence.
	- Mitra's f.g. counterexample shows the finite-presentedness of $G$ cannot be dropped: the escaping short loops are carried by the HNN relators $tu_it^{-1}v_i^{-1}$, which force no finite 2-complex model.

	I could not push the reformulation to a full proof: for hyperbolic $G$, distortion of $H$ means short $b_n\in G$ with $b_n\in H$, $\|b_n\|_H\to\infty$, and the question becomes whether such short $H$-elements must reappear (as a fixed $b$, or with centralizers) in unboundedly many conjugates of $H$. Hyperbolicity makes individual conjugates $zbz^{-1}$ long, so a positive answer requires controlling how the family of short distorted elements distributes across cosets — precisely the content of a uniform finite-height theorem for arbitrary (possibly distorted) f.p. subgroups, which is not in the literature I found.

	## What remains

	- The question is open for finitely presented $G$, and specifically for $G$ word-hyperbolic; also open: the weaker variant asking only whether $I(x)\to\infty$ implies $H$ is hyperbolic / admits a Cannon–Thurston map.
	- Via the reformulation, a positive answer for hyperbolic $G$ is equivalent to: distortion of $H\le G$ forces a fixed conjugacy class (or a centralizer) to meet unboundedly many conjugates of $H$. A promising route is to combine the annular-diagram structure of conjugacy in hyperbolic groups with the Gitik–Mitra–Rips–Sageev width theory; the obstacle is that distorted subgroups need not have finite width, and no counterexample with $I(x)\to\infty$ and distorted $H$ is known either.
	- Honest caveats: (i) I did not verify the current status of Q 1.8 (Swarup) beyond Mitra's 2004 partial answer — a full resolution of Q 1.8 would likely interact with Q 1.9; (ii) the Cannon–Thurston–Peano-curve reference (Geom. Topol. 11 (2007) 1315–1355) was seen only in reference lists, not independently Crossref-checked; (iii) absence of a resolution in the literature is established by search, not by any systematic review — a negative (counterexample) answer could exist in sources I did not reach.