Datasets:

ulamai
/

UnsolvedMath

	---
	id: AMR-010-0106
	classification: OPEN-TRIAGE
	wording_corrected: no
	---

	# AMR-010-0106 — Gromov's surface subgroup question for one-ended hyperbolic groups

	## Problem (corrected statement if needed)

	Source: Mladen Bestvina, Questions in Geometric Group Theory (author-hosted PDF, major revision Aug 2000, updated July 2004), https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf, Question 1.6 (§1.2 "Subgroups of Hyperbolic Groups"). Original wording, fetched and checked verbatim:

	> Q 1.6. (Gromov) Does every 1-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?

	The dataset transcription is exact — no correction needed. A "closed hyperbolic surface subgroup" means a subgroup isomorphic to π₁(S) for S a closed surface of genus ≥ 2. The one-ended hypothesis rules out the degenerate cases: finite groups (0 ends), virtually cyclic groups (2 ends), and nontrivial free products / splittings over finite groups (∞ ends), all of which can be word-hyperbolic without containing a closed surface group (a free group contains none, since every subgroup of a free group is free, and a closed surface group is not free; the same holds for free products of finite groups by Kurosh). Bestvina notes the question is "inspired by the well-known conjecture that closed aspherical 3-manifolds are virtually [Haken]".

	## Status / Literature

	Open in general as of this review (checked August 2026). The general case remains unresolved, but there is a rich body of positive partial results and a meaningful reduction. All citations below were verified against Crossref or the arXiv API during this review.

	- Kahn–Markovic 2012 — Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012), 1127–1190. DOI: [10.4007/annals.2012.175.3.4](https://doi.org/10.4007/annals.2012.175.3.4) (verified via Crossref). Establishes the Surface Subgroup Theorem: every closed hyperbolic 3-manifold group contains a (quasiconvex, in fact immersed almost-geodesic) closed surface subgroup. This is the motivating special case of Gromov's question. (The cusped/finite-volume case is also known, by work of Masters–Zhang and of Baker–Cooper; I did not independently verify those DOIs, so I flag them as unverified here.)

	- Gordon–Long–Reid 2004 — Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004), 135–148. DOI: [10.1016/j.jpaa.2003.10.011](https://doi.org/10.1016/j.jpaa.2003.10.011) (verified via Crossref reference record). Surface subgroups in certain hyperbolic Coxeter and Artin groups.

	- Calegari 2008 — Surface subgroups from homology, Geom. Topol. 12 (2008), 1995–2007. DOI: [10.2140/gt.2008.12.1995](https://doi.org/10.2140/gt.2008.12.1995) (verified via Crossref reference record). Proves that graphs of free groups amalgamated over cyclic subgroups contain surface subgroups under a homological hypothesis, using stable commutator length.

	- Kim–Oum 2014 — Hyperbolic surface subgroups of one-ended doubles of free groups, J. Topol. 7 (2014), 927–947. DOI: [10.1112/jtopol/jtu004](https://doi.org/10.1112/jtopol/jtu004) (verified via Crossref). Positive answer for one-ended doubles F *_w F when rank(F) = 2, or when the amalgamating words use every generator equally often; the abstract explicitly frames the work as attacking Gromov's question.

	- Calegari–Walker 2015 — Random groups contain surface subgroups, J. Amer. Math. Soc. 28 (2015), 383–419. arXiv:[1304.2188](https://arxiv.org/abs/1304.2188) (verified via arXiv API; journal DOI 10.1090/S0894-0347-2014-00802-X appears in Crossref records). In Gromov's few-relators/density models, a random group — which is one-ended and hyperbolic with probability → 1 in the appropriate range — contains many quasiconvex surface subgroups. So the question has a positive answer for "generic" hyperbolic groups.

	- Wilton 2018 — Essential surfaces in graph pairs, J. Amer. Math. Soc. 31 (2018), 893–919. DOI: [10.1090/jams/901](https://doi.org/10.1090/jams/901) (verified via Crossref, including the abstract). The strongest structural result to date: a positive answer whenever Γ is the fundamental group of a graph of free groups with cyclic edge groups, and, crucially, a reduction of Gromov's question: every one-ended hyperbolic group without 2-torsion contains either a quasiconvex surface subgroup or a quasiconvex rigid subgroup (one that does not split over a virtually cyclic subgroup). Hence, modulo the 2-torsion assumption, it suffices to resolve the question for rigid hyperbolic groups. The same paper also finds surface subgroups in limit groups.

	- Markovic 2013 — Criterion for Cannon's conjecture, Geom. Funct. Anal. 23 (2013), 1035–1061. DOI: [10.1007/s00039-013-0228-5](https://doi.org/10.1007/s00039-013-0228-5) (verified via Crossref reference record). Surveys the problem (as "Problem 1.1 (Gromov)") and links it to Cannon's conjecture: a positive answer for groups with S² boundary, plus a quasiconvexity statement, would give a criterion for a hyperbolic group to be Kleinian.

	- Ng 2025 — Quasi-convex surface subgroups in some one-relator groups with torsion, arXiv:[2510.01876](https://arxiv.org/abs/2510.01876) (verified via arXiv API; v2, June 2026). Its introduction (October 2025) describes Gromov's question as still open ("A longstanding question often attributed to Gromov asks whether every one-ended hyperbolic group contains a ... surface subgroup ... This has generated a lot of work"), confirming no full solution had appeared as of late 2025; it also cites Wilton's reduction as the current state of the art.

	## Work done

	1. Fetched the Bestvina questions PDF directly and confirmed the dataset transcription of Q 1.6 character-for-character (including the "(Gromov)" attribution and the surrounding §1.2 context).
	2. Searched the web for the current status, for claims of a full solution or counterexample (none found), and for recent activity.
	3. Verified every cited publication against Crossref (`api.crossref.org/works/...`) or the arXiv API; bibliographic details above come from those records, not from memory. Two sources I did not verify (Masters–Zhang, Baker–Cooper) are explicitly flagged as unverified.
	4. Considered whether a direct attack is feasible within this review's scope: it is not. The problem has resisted Gromov's school and two decades of geometric group theory; even the strongest known general result (Wilton's reduction) required new machinery (cycle precursors, essential surfaces in graph pairs).

	Mathematical reasoning about the shape of the problem (why the hypotheses are right, and where the difficulty lies):

	- One-endedness is necessary and essentially sharp. Any hyperbolic group admits a Dunwoody–Stallings splitting as a graph of groups with finite edge groups and vertex groups that are finite or one-ended; closed surface subgroups, being one-ended themselves, must lie (up to conjugacy) in one-ended vertex groups. So the question genuinely reduces to the one-ended case, and the hypothesis cannot be weakened.
	- JSJ decomposition reduces further. A one-ended hyperbolic group that splits over a 2-ended (virtually cyclic) subgroup has a JSJ decomposition; if all the pieces were handled (they are, when the pieces are free or surface-type), the combination problem remains — and this is exactly what Calegari, Kim–Oum, and Wilton attack. Wilton's theorem completes this line for groups without 2-torsion: either the splitting data already yields a quasiconvex surface subgroup, or the group contains a quasiconvex rigid subgroup. The residual core problem is therefore: does every rigid one-ended hyperbolic group (without 2-torsion) contain a surface subgroup?
	- Why the rigid case is hard. Rigid hyperbolic groups include fundamental groups of closed negatively curved manifolds in dimension ≥ 4 (where the Kahn–Markovic "good pants" machinery, which depends on the 2-dimensional geometry of immersed surfaces in 3-manifolds and on exponential mixing of the frame flow, does not apply), lattices in other rank-one groups (e.g. quaternionic hyperbolic lattices — many of which have property (T)-like rigidity phenomena), and Gromov–Kapovich–Kleiner-type groups with exotic boundaries (e.g. the Menger curve or Sierpiński carpet), which are frequently cited as candidate counterexamples. There is no known obstruction, but also no general construction.
	- Consequences worth noting. A positive answer would, combined with residual finiteness (itself a famous open problem, Q 1.15 on the same list), have structural consequences; Bridson–Conder–Reid (Israel J. Math. 214, 2016; seen in search results, not independently DOI-verified) show that if every one-ended hyperbolic group were residually finite and contained a quasiconvex surface subgroup, then certain embeddings T ≥ F with T one-ended hyperbolic and F free would be impossible.

	## Result

	OPEN-TRIAGE. The problem is a famous open question of Gromov, transcribed correctly from Bestvina's list, and it remains unsolved in full generality as of August 2026. The literature state is:

	- Solved cases: closed hyperbolic 3-manifold groups (Kahn–Markovic), graphs of free groups with cyclic edge groups (Wilton, extending Calegari and Kim–Oum for doubles), limit groups (Wilton), random groups (Calegari–Walker), various Coxeter/Artin and one-relator families (Gordon–Long–Reid; Ng 2025).
	- General reduction (Wilton 2018): without 2-torsion, the question reduces to rigid one-ended hyperbolic groups — those with no splitting over virtually cyclic subgroups.
	- No counterexample is known, and no known obstruction exists; the generic case is positive.

	## What remains

	1. The rigid case: prove or disprove that every rigid one-ended hyperbolic group contains a closed surface subgroup. Sub-cases of particular interest: closed negatively curved manifolds of dimension ≥ 4 (not known to contain immersed surfaces in general), Kapovich–Kleiner and related boundary-exotic hyperbolic groups (candidate counterexamples), and rigid small-cancellation groups.
	2. Remove the 2-torsion hypothesis in Wilton's reduction (currently a technical gap: the reduction is proved only for groups without 2-torsion).
	3. The quasiconvex strengthening (often called Q (A′) in the literature): does every one-ended hyperbolic group contain a quasiconvex surface subgroup? Even where surface subgroups are known, quasiconvexity is not always established.
	4. Interaction with other open questions on the same list: residual finiteness of hyperbolic groups (Q 1.15) and Cannon's conjecture (cf. Q 1.18 remarks and Markovic's criterion) — a positive answer to the surface subgroup question for S²-boundary groups would be a key input.
	5. Natural next steps for a researcher: attempt the rigid case for specific families (e.g. rigid one-relator groups, building on Ng's 2025 work; or 4-dimensional hyperbolic manifolds via new immersed-surface constructions), or seek a counterexample among rigid groups with Menger-curve boundary.

	---
	id: AMR-010-0106
	classification: OPEN-TRIAGE
	wording_corrected: no
	---

	# AMR-010-0106 — Gromov's surface subgroup question for one-ended hyperbolic groups

	## Problem (corrected statement if needed)

	Source: Mladen Bestvina, Questions in Geometric Group Theory (author-hosted PDF, major revision Aug 2000, updated July 2004), https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf, Question 1.6 (§1.2 "Subgroups of Hyperbolic Groups"). Original wording, fetched and checked verbatim:

	> Q 1.6. (Gromov) Does every 1-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?

	The dataset transcription is exact — no correction needed. A "closed hyperbolic surface subgroup" means a subgroup isomorphic to π₁(S) for S a closed surface of genus ≥ 2. The one-ended hypothesis rules out the degenerate cases: finite groups (0 ends), virtually cyclic groups (2 ends), and nontrivial free products / splittings over finite groups (∞ ends), all of which can be word-hyperbolic without containing a closed surface group (a free group contains none, since every subgroup of a free group is free, and a closed surface group is not free; the same holds for free products of finite groups by Kurosh). Bestvina notes the question is "inspired by the well-known conjecture that closed aspherical 3-manifolds are virtually [Haken]".

	## Status / Literature

	Open in general as of this review (checked August 2026). The general case remains unresolved, but there is a rich body of positive partial results and a meaningful reduction. All citations below were verified against Crossref or the arXiv API during this review.

	- Kahn–Markovic 2012 — Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012), 1127–1190. DOI: [10.4007/annals.2012.175.3.4](https://doi.org/10.4007/annals.2012.175.3.4) (verified via Crossref). Establishes the Surface Subgroup Theorem: every closed hyperbolic 3-manifold group contains a (quasiconvex, in fact immersed almost-geodesic) closed surface subgroup. This is the motivating special case of Gromov's question. (The cusped/finite-volume case is also known, by work of Masters–Zhang and of Baker–Cooper; I did not independently verify those DOIs, so I flag them as unverified here.)

	- Gordon–Long–Reid 2004 — Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004), 135–148. DOI: [10.1016/j.jpaa.2003.10.011](https://doi.org/10.1016/j.jpaa.2003.10.011) (verified via Crossref reference record). Surface subgroups in certain hyperbolic Coxeter and Artin groups.

	- Calegari 2008 — Surface subgroups from homology, Geom. Topol. 12 (2008), 1995–2007. DOI: [10.2140/gt.2008.12.1995](https://doi.org/10.2140/gt.2008.12.1995) (verified via Crossref reference record). Proves that graphs of free groups amalgamated over cyclic subgroups contain surface subgroups under a homological hypothesis, using stable commutator length.

	- Kim–Oum 2014 — Hyperbolic surface subgroups of one-ended doubles of free groups, J. Topol. 7 (2014), 927–947. DOI: [10.1112/jtopol/jtu004](https://doi.org/10.1112/jtopol/jtu004) (verified via Crossref). Positive answer for one-ended doubles F *_w F when rank(F) = 2, or when the amalgamating words use every generator equally often; the abstract explicitly frames the work as attacking Gromov's question.

	- Calegari–Walker 2015 — Random groups contain surface subgroups, J. Amer. Math. Soc. 28 (2015), 383–419. arXiv:[1304.2188](https://arxiv.org/abs/1304.2188) (verified via arXiv API; journal DOI 10.1090/S0894-0347-2014-00802-X appears in Crossref records). In Gromov's few-relators/density models, a random group — which is one-ended and hyperbolic with probability → 1 in the appropriate range — contains many quasiconvex surface subgroups. So the question has a positive answer for "generic" hyperbolic groups.

	- Wilton 2018 — Essential surfaces in graph pairs, J. Amer. Math. Soc. 31 (2018), 893–919. DOI: [10.1090/jams/901](https://doi.org/10.1090/jams/901) (verified via Crossref, including the abstract). The strongest structural result to date: a positive answer whenever Γ is the fundamental group of a graph of free groups with cyclic edge groups, and, crucially, a reduction of Gromov's question: every one-ended hyperbolic group without 2-torsion contains either a quasiconvex surface subgroup or a quasiconvex rigid subgroup (one that does not split over a virtually cyclic subgroup). Hence, modulo the 2-torsion assumption, it suffices to resolve the question for rigid hyperbolic groups. The same paper also finds surface subgroups in limit groups.

	- Markovic 2013 — Criterion for Cannon's conjecture, Geom. Funct. Anal. 23 (2013), 1035–1061. DOI: [10.1007/s00039-013-0228-5](https://doi.org/10.1007/s00039-013-0228-5) (verified via Crossref reference record). Surveys the problem (as "Problem 1.1 (Gromov)") and links it to Cannon's conjecture: a positive answer for groups with S² boundary, plus a quasiconvexity statement, would give a criterion for a hyperbolic group to be Kleinian.

	- Ng 2025 — Quasi-convex surface subgroups in some one-relator groups with torsion, arXiv:[2510.01876](https://arxiv.org/abs/2510.01876) (verified via arXiv API; v2, June 2026). Its introduction (October 2025) describes Gromov's question as still open ("A longstanding question often attributed to Gromov asks whether every one-ended hyperbolic group contains a ... surface subgroup ... This has generated a lot of work"), confirming no full solution had appeared as of late 2025; it also cites Wilton's reduction as the current state of the art.

	## Work done

	1. Fetched the Bestvina questions PDF directly and confirmed the dataset transcription of Q 1.6 character-for-character (including the "(Gromov)" attribution and the surrounding §1.2 context).
	2. Searched the web for the current status, for claims of a full solution or counterexample (none found), and for recent activity.
	3. Verified every cited publication against Crossref (`api.crossref.org/works/...`) or the arXiv API; bibliographic details above come from those records, not from memory. Two sources I did not verify (Masters–Zhang, Baker–Cooper) are explicitly flagged as unverified.
	4. Considered whether a direct attack is feasible within this review's scope: it is not. The problem has resisted Gromov's school and two decades of geometric group theory; even the strongest known general result (Wilton's reduction) required new machinery (cycle precursors, essential surfaces in graph pairs).

	Mathematical reasoning about the shape of the problem (why the hypotheses are right, and where the difficulty lies):

	- One-endedness is necessary and essentially sharp. Any hyperbolic group admits a Dunwoody–Stallings splitting as a graph of groups with finite edge groups and vertex groups that are finite or one-ended; closed surface subgroups, being one-ended themselves, must lie (up to conjugacy) in one-ended vertex groups. So the question genuinely reduces to the one-ended case, and the hypothesis cannot be weakened.
	- JSJ decomposition reduces further. A one-ended hyperbolic group that splits over a 2-ended (virtually cyclic) subgroup has a JSJ decomposition; if all the pieces were handled (they are, when the pieces are free or surface-type), the combination problem remains — and this is exactly what Calegari, Kim–Oum, and Wilton attack. Wilton's theorem completes this line for groups without 2-torsion: either the splitting data already yields a quasiconvex surface subgroup, or the group contains a quasiconvex rigid subgroup. The residual core problem is therefore: does every rigid one-ended hyperbolic group (without 2-torsion) contain a surface subgroup?
	- Why the rigid case is hard. Rigid hyperbolic groups include fundamental groups of closed negatively curved manifolds in dimension ≥ 4 (where the Kahn–Markovic "good pants" machinery, which depends on the 2-dimensional geometry of immersed surfaces in 3-manifolds and on exponential mixing of the frame flow, does not apply), lattices in other rank-one groups (e.g. quaternionic hyperbolic lattices — many of which have property (T)-like rigidity phenomena), and Gromov–Kapovich–Kleiner-type groups with exotic boundaries (e.g. the Menger curve or Sierpiński carpet), which are frequently cited as candidate counterexamples. There is no known obstruction, but also no general construction.
	- Consequences worth noting. A positive answer would, combined with residual finiteness (itself a famous open problem, Q 1.15 on the same list), have structural consequences; Bridson–Conder–Reid (Israel J. Math. 214, 2016; seen in search results, not independently DOI-verified) show that if every one-ended hyperbolic group were residually finite and contained a quasiconvex surface subgroup, then certain embeddings T ≥ F with T one-ended hyperbolic and F free would be impossible.

	## Result

	OPEN-TRIAGE. The problem is a famous open question of Gromov, transcribed correctly from Bestvina's list, and it remains unsolved in full generality as of August 2026. The literature state is:

	- Solved cases: closed hyperbolic 3-manifold groups (Kahn–Markovic), graphs of free groups with cyclic edge groups (Wilton, extending Calegari and Kim–Oum for doubles), limit groups (Wilton), random groups (Calegari–Walker), various Coxeter/Artin and one-relator families (Gordon–Long–Reid; Ng 2025).
	- General reduction (Wilton 2018): without 2-torsion, the question reduces to rigid one-ended hyperbolic groups — those with no splitting over virtually cyclic subgroups.
	- No counterexample is known, and no known obstruction exists; the generic case is positive.

	## What remains

	1. The rigid case: prove or disprove that every rigid one-ended hyperbolic group contains a closed surface subgroup. Sub-cases of particular interest: closed negatively curved manifolds of dimension ≥ 4 (not known to contain immersed surfaces in general), Kapovich–Kleiner and related boundary-exotic hyperbolic groups (candidate counterexamples), and rigid small-cancellation groups.
	2. Remove the 2-torsion hypothesis in Wilton's reduction (currently a technical gap: the reduction is proved only for groups without 2-torsion).
	3. The quasiconvex strengthening (often called Q (A′) in the literature): does every one-ended hyperbolic group contain a quasiconvex surface subgroup? Even where surface subgroups are known, quasiconvexity is not always established.
	4. Interaction with other open questions on the same list: residual finiteness of hyperbolic groups (Q 1.15) and Cannon's conjecture (cf. Q 1.18 remarks and Markovic's criterion) — a positive answer to the surface subgroup question for S²-boundary groups would be a key input.
	5. Natural next steps for a researcher: attempt the rigid case for specific families (e.g. rigid one-relator groups, building on Ng's 2025 work; or 4-dimensional hyperbolic manifolds via new immersed-surface constructions), or seek a counterexample among rigid groups with Menger-curve boundary.