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| id: AMR-010-0113 |
| classification: SOLVED-IN-LITERATURE |
| wording_corrected: no |
| --- |
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| # AMR-010-0113 — Combination theorem for relatively hyperbolic groups (Swarup) |
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| ## Problem (corrected statement if needed) |
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| Source: M. Bestvina, "Questions in Geometric Group Theory" (major revision August 2000, updated July 2004), Question 1.13 (PDF page 3), author-hosted PDF at https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf . |
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| Original wording, verified against the source PDF: |
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| > **Q 1.13. (Swarup)** Prove the combination theorem for relatively hyperbolic groups. |
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| The July-2004 update in the list itself already records the solution: |
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| > "Update: Dahmani [Dah03] and Alibegović [Ali] have versions adapted for use to limit groups." |
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| The dataset transcription matches the source exactly; no correction needed. The question asks for the relative analogue of the Bestvina–Feighn Combination Theorem for hyperbolic groups (M. Bestvina and M. Feighn, "A combination theorem for negatively curved groups", *J. Differential Geom.* 35 (1992), 85–101, DOI 10.4310/jdg/1214447806). |
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| ## Status / Literature |
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| **Solved in the literature, by three complementary theorems** (all citations verified via Crossref / arXiv): |
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| 1. **F. Dahmani, "Combination of convergence groups", *Geometry & Topology* 7 (2003), 933–963.** DOI 10.2140/gt.2003.7.933 (verified on Crossref). A *dynamical* combination theorem: he shows that suitable amalgamated products / HNN extensions of relatively hyperbolic groups (viewed as convergence groups) are again relatively hyperbolic, with the expected peripheral structure. This was the first published combination theorem for relative hyperbolicity and was tailored to applications to limit groups — it is the key tool proving that limit groups are relatively hyperbolic with respect to their maximal abelian subgroups of rank ≥ 2 (Q 3.8 of the same list). |
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| 2. **E. Alibegović, "A combination theorem for relatively hyperbolic groups", *Bulletin of the London Mathematical Society* 37(3) (2005), 459–466.** DOI 10.1112/S0024609304004059 (verified on Crossref); arXiv:math/0310257 (abstract page verified). A Bestvina–Feighn-style combination theorem for amalgams of relatively hyperbolic groups along "liminal" edge groups satisfying malnormality-type conditions, again with applications to limit groups. |
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| 3. **M. Mj and L. Reeves, "A combination theorem for strong relative hyperbolicity", *Geometry & Topology* 12(3) (2008), 1777–1798.** DOI 10.2140/gt.2008.12.1777 (verified via Crossref reference lists of later papers and the Project Euclid page); arXiv:math/0611601 (abstract verified — the abstract states explicitly: "This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and **answers a question of Swarup**."). A *geometric* combination theorem for trees of (strongly) relatively hyperbolic metric spaces, with conditions different from those of Dahmani and Alibegović, plus a converse to the main theorem. |
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| Later refinements/variants also exist, e.g. R. Tomar, "A combination theorem for relatively acylindrical graphs of relatively hyperbolic groups", *Topology Appl.* 380 (2026), 109692, DOI 10.1016/j.topol.2025.109692 (verified on Crossref), and an unpublished algebraic version for 2-complexes of relatively hyperbolic groups by F. Gautero ("An algebraic combination theorem for graphs of relatively hyperbolic groups", preprint, 2011, author-hosted; not formally published, cited but not verified as refereed). |
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| ## Work done |
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| - Read `worklist/AMR-010-0113.md`; fetched the Bestvina source PDF and confirmed the exact wording of Q 1.13 and the July-2004 update line. |
| - Web-searched for the resolution; identified the three main papers above. |
| - Verified each citation against Crossref (`api.crossref.org/works/...`): Dahmani (DOI 10.2140/gt.2003.7.933 — full record including title, journal, volume, pages), Alibegović (DOI 10.1112/S0024609304004059 — full record), Mj–Reeves (DOI 10.2140/gt.2008.12.1777 — confirmed via Project Euclid listing and via the Crossref-deposited reference lists of Krishna, *Proc. Math. Sci.* 130 (2020), and Tomar 2026), Bestvina–Feighn (DOI 10.4310/jdg/1214447806 — confirmed in Crossref reference lists). arXiv abstract pages for math/0611601 and math/0310257 were fetched and confirm titles/authors. |
| - No independent new mathematics was attempted: the question is fully settled in the published literature, so the appropriate output is a rigorous triage (a "solve-by-you" attempt at an L3 problem already solved by three major papers would add nothing). |
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| ## Result |
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| Swarup's question is **answered affirmatively in the literature**, in three distinct frameworks that mirror the different definitions of relative hyperbolicity: |
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| - **Dynamical/convergence-group version (Dahmani 2003).** If groups acting as convergence groups (relatively hyperbolic) are combined along parabolic-type ("liminal") subgroups satisfying geometric-finiteness and intersection-control hypotheses, the amalgamated product/HNN extension acts as a convergence group on a suitably assembled compactum and is relatively hyperbolic relative to the expected peripherals. Applied to show limit groups are relatively hyperbolic w.r.t. maximal noncyclic abelian subgroups. |
| - **Amalgam version (Alibegović 2005).** For a one-edge graph of relatively hyperbolic groups with liminal edge group satisfying an almost-malnormality condition and a compatibility ("isolated"-type) condition on the peripherals, the fundamental group of the graph of groups is relatively hyperbolic relative to the images of the vertex peripherals not contained in the edge group. |
| - **Geometric version (Mj–Reeves 2008), the one explicitly billed as answering Swarup's question.** For a tree of strongly relatively hyperbolic metric spaces satisfying (i) the qi-embedded condition (edge spaces quasi-isometrically embed into vertex spaces with edge-to-vertex qi-embeddings), (ii) the strictly type-preserving condition (peripheral/horosphere-like sets map to peripheral sets), and (iii) a uniform hallway-flare condition (the relative analogue of the Bestvina–Feighn flare condition), the total space is strongly relatively hyperbolic relative to the natural family of horosphere-like subsets; a **weak** combination theorem (electrocution/electric-space hyperbolicity) holds under (i)+(ii) with a milder flare condition, and they prove a **converse**: strong relative hyperbolicity of the total space forces the qi-embedded condition. |
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| Together these subsume the classical Bestvina–Feighn theorem (recover it by taking all peripherals trivial/hyperbolic) and establish the general combination principle Swarup asked for. |
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| ## What remains |
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| The original question is closed. Remaining activity is in refinements rather than in the problem itself: |
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| - Combination theorems under weaker hypotheses (e.g. relatively acylindrical splittings; Tomar 2026; Pal–Tomar work on finite relative height of splittings, arXiv:2207.03167 — not fully verified here). |
| - A fully published algebraic combination theorem for general 2-complexes of relatively hyperbolic groups (Gautero's version remains a preprint). |
| - Companion questions: Cannon–Thurston maps and limit-set intersection theorems for the combined relatively hyperbolic group (partially answered by Mj–Pal, Sardar, Krishna). |
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