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| id: AMR-010-0112 |
| classification: SOLVED-IN-LITERATURE |
| wording_corrected: no |
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| # AMR-010-0112 — Whyte's question: finite-index subgroups isomorphic to infinite-index subgroups in 1-ended hyperbolic groups |
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| ## Problem (corrected statement if needed) |
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| The dataset transcription is faithful to the source. The problem is Question 1.12 in Mladen Bestvina's open-problem list "Questions in Geometric Group Theory" (major revision August 22, 2000; updated version at https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf), attributed to Kevin Whyte: |
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| > (Whyte) Let Γ be a 1-ended hyperbolic group. Can a finite index subgroup of Γ be isomorphic to a subgroup of Γ of infinite index? |
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| No correction needed. Note the attribution caveat in Bestvina's list: "Names in parentheses reflect the person I heard the question from." The same question was also asked by Kapovich (I. Kapovich, "Arithmetic aspects of self-similar groups", Groups Geom. Dyn. 6 (2012), DOI: 10.4171/GGD/172, Section 2, where the property of admitting no such pair is called "weakly coHopfian"). |
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| ## Status / Literature |
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| **Answer: YES — such groups exist. The question is fully resolved (affirmatively) by Stark–Woodhouse (2021).** |
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| Verified citations: |
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| 1. **E. Stark, D. J. Woodhouse, "Hyperbolic Groups That Are Not Commensurably Co-Hopfian"**, International Mathematics Research Notices (IMRN) 2021, no. 1, 579–595. DOI: 10.1093/imrn/rnaa033 (verified via Crossref API); arXiv:1812.07799 (verified via arXiv API; v3, 2020). |
| - A group Γ is *commensurably coHopfian* if no finite-index subgroup of Γ is isomorphic to an infinite-index subgroup of Γ. Whyte's question asks whether every 1-ended hyperbolic group is commensurably coHopfian. The paper explicitly states it answers Whyte's Question 1.12 from Bestvina's list. |
| - **Theorem 1.1:** There exist one-ended hyperbolic groups that are not commensurably coHopfian. Main example: a *simple surface amalgam* X built from three genus-one surfaces with one boundary component, boundaries identified. They construct a degree-3 cover X₁ → X (each surface covered by a genus-2 one-boundary-component surface) and a degree-4 cover X₂ → X, with a π₁-injective proper embedding X₁ ↪ X₂ (a retraction). Then π₁(X₁) ≅ π₁(X₂) sit inside π₁(X): the former has finite index (degree-4 cover), the latter infinite index, yet π₁(X₁) embeds in π₁(X₂), so π₁(X) contains a finite-index subgroup isomorphic to an infinite-index subgroup. |
| - **Theorem 1.2:** The fundamental group of *every* simple surface amalgam (union of k ≥ 3 negative-Euler-characteristic one-boundary-component surfaces with boundaries identified) is not commensurably coHopfian. These groups are one-ended and hyperbolic (Bestvina–Feighn combination theorem; they even admit CAT(-1) metrics). |
| - Context within the paper: Sela proved every torsion-free one-ended hyperbolic group is coHopfian (Moioli's thesis extended this to all one-ended hyperbolic groups), so the answer to Whyte's question was genuinely uncertain; Strebel (Comment. Math. Helv. 52 (1977), DOI: 10.1007/BF02567371) proved infinite-index subgroups of Poincaré duality groups have strictly smaller cohomological dimension, hence PD groups (e.g. closed hyperbolic manifold groups) ARE commensurably coHopfian — so the answer is "yes in general, no for PD groups". |
| - The constructed infinite-index embeddings are retractions, hence quasi-isometric embeddings that are not quasi-isometries; these are also the first known examples of one-ended hyperbolic groups that are not quasi-isometrically coHopfian. |
| - The paper poses Conjecture 1.3: failure of commensurable coHopficity for a one-ended hyperbolic group should be tied to the presence of maximal hanging Fuchsian vertex groups in the Bowditch JSJ decomposition. |
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| 2. **N. Lazarovich, "Finite index rigidity of hyperbolic groups"**, arXiv:2302.04484 (v3, 2024; verified via arXiv API). |
| - Proves that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index. This settles (negatively) the closely related follow-up question recorded as Question 1.6 in the Stark–Woodhouse paper (attributed to Bestvina): no one-ended hyperbolic group contains isomorphic finite-index subgroups of *different* indices. It complements Stark–Woodhouse: the finite-index/infinite-index phenomenon cannot occur between two finite-index subgroups. |
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| ## Work done |
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| - Read the dataset item and identified the source as Bestvina's "Questions in Geometric Group Theory", Q1.12 (Whyte). The wording matches the author's PDF verbatim, so no correction was needed. |
| - Web-searched the question; located the Oxford ORA preprint and the arXiv listing of the Stark–Woodhouse paper, which explicitly states it answers Whyte's Question 1.12 on Bestvina's list. |
| - Fetched and read the Stark–Woodhouse preprint (introduction and the main construction in Sections 2–3) to confirm exactly what is proved and how. |
| - Verified the publication record via Crossref (DOI 10.1093/imrn/rnaa033, IMRN 2021(1), 579–595) and the arXiv API (arXiv:1812.07799; Lazarovich arXiv:2302.04484). All citations in this report were verified to exist through one of these two APIs or the Bestvina PDF itself. |
| - Reproduced the logical structure of the main example independently (see Result) — the construction is elementary (covering-space theory plus Euler characteristic bookkeeping via Neumann's Lemma 3.2 in Algebr. Geom. Topol. 1 (2001), DOI: 10.2140/agt.2001.1.411). |
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| ## Result |
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| The question is solved in the literature with answer **yes**: there exist one-ended hyperbolic groups Γ containing a finite-index subgroup H ≤ Γ and an infinite-index subgroup K ≤ Γ with H ≅ K. |
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| Sketch of the Stark–Woodhouse main example (my summary of their §2): Let X = Σ₁ ∪ Σ₂ ∪ Σ₃ where each Σᵢ is a genus-1 surface with one boundary circle, all boundaries glued to a single S¹. Then π₁(X) is a one-ended hyperbolic group (Bestvina–Feighn, since each π₁(Σᵢ) is free amalgamated along a malnormal cyclic subgroup). |
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| - *Degree-3 cover X₁:* by Neumann's covering lemma, each Σᵢ has a 3-sheeted cover with exactly one boundary component; its Euler characteristic is 3·(−1) = −3, so it is a genus-2 surface with one boundary. Gluing gives a degree-3 cover X₁ → X of the same "simple surface amalgam" form. |
| - *Degree-4 cover X₂:* each Σᵢ has a 2-sheeted cover Σᵢ″ with two boundary components (still genus 1). Glue one boundary component of each Σᵢ″ to form one amalgam circle, and attach extra copies of the Σⱼ along the other boundary components; this gives a degree-4 cover X₂ → X. |
| - *Embedding:* X₁ embeds π₁-injectively as a proper sub-amalgam of X₂ (visibly a retract of X₂), so π₁(X₁) ≅ π₁(X₂) appears inside π₁(X₂) as an infinite-index subgroup. Since both π₁(X₁) (index 3) and π₁(X₂) (index 4) are finite-index in π₁(X), the group Γ = π₁(X) contains a finite-index subgroup (π₁(X₁), via index 3) isomorphic to an infinite-index subgroup (π₁(X₁) ⊂ π₁(X₂) ⊂ Γ). ∎ |
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| Their Theorem 3.1 extends this to all simple surface amalgams by solving a linear system in covering degrees (their Claim 3.2) to build two finite covers X′, X″ with X′ embedding π₁-injectively in X″. |
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| The phenomenon is genuinely new relative to classical rigidity: one-ended hyperbolic groups are coHopfian (Sela; Moioli), and Poincaré duality groups (e.g. closed hyperbolic manifold groups) are commensurably coHopfian by Strebel's cohomological-dimension argument — so Whyte's question has answer "yes" in general but "no" for important subclasses. |
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| ## What remains |
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| - **Characterization problem (Stark–Woodhouse Conjecture 1.3, still open to my knowledge):** for a one-ended hyperbolic group, is failure of commensurable coHopficity equivalent to the presence of maximal hanging Fuchsian vertex groups in the Bowditch JSJ decomposition? Stark–Woodhouse give supporting examples on both sides (mixed JSJ examples that are and are not commensurably coHopfian) but the general conjecture is open; they caution that highly distorted (non-quasiconvex) embeddings may require a quasiconvexity hypothesis. |
| - **Quasi-isometric coHopficity:** the embeddings constructed are retractions, hence the first examples of one-ended hyperbolic groups failing to be quasi-isometrically coHopfian. Classifying which one-ended hyperbolic groups are QI-coHopfian remains open (related work: Kapovich–Lukyanenko for non-uniform rank-one lattices, DOI: 10.1090/S1088-4173-2012-00246-9). |
| - **Different indices, both finite:** Lazarovich (arXiv:2302.04484) closed the variant asking for isomorphic finite-index subgroups of *different* indices — impossible for non-elementary hyperbolic groups. |
| - Whether the commensurable-coHopficity dichotomy can be detected from the Gromov boundary or conformal dimension appears unexplored. |
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