id: AMR-010-0108
classification: OPEN-TRIAGE
wording_corrected: 'no'
AMR-010-0108 — Swarup's question: is a finitely presented, almost-normal, finite-height subgroup of a hyperbolic group quasiconvex?
Problem (corrected statement if needed)
The dataset transcription is faithful to the source: Question 1.8 of M. Bestvina's
problem list Questions in Geometric Group Theory (major revision August 2000; the
questions-updated.pdf version accessed for the dataset), attributed to G. A. Swarup:
Q 1.8 (Swarup). Suppose $H$ is a finitely presented subgroup of a word-hyperbolic group $G$ which has finite index in its normalizer. Assume that there is $n>0$ such that the intersection of $n$ distinct conjugates of $H$ is always finite. Is $H$ quasi-convex in $G$?
The list itself adds: "The converse is a theorem of [Gitik–Mitra–Rips–Sageev]. A special case worth considering is when $G$ splits over $H$ when Gersten's converse of the combination theorem might be helpful. Remark (Gitik): The problem is open even when $H$ is malnormal in $G$." (Source PDF: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf — the wording above was checked against the search-indexed text of the PDF; the PDF has the typo "Swarvp".)
One interpretive note: "distinct conjugates" should be read as essentially distinct conjugates in the sense of Gitik–Mitra–Rips–Sageev (conjugates by elements in distinct cosets of $H$, or equivalently — since $[N_G(H):H]<\infty$ here — by elements in distinct cosets of $N_G(H)$). With this reading, the hypothesis "$n$ distinct conjugates always have finite intersection" says exactly that $H$ has height $\le n-1$ in $G$ in the GMRS sense. Because $[N_G(H):H]<\infty$, distinct and essentially distinct conjugates differ only by the bounded factor $[N_G(H):H]$, so the two formulations of the hypothesis are equivalent.
Status / Literature
Open as of early 2026 — see the 2025 preprint of Halder–Sardar below, which states explicitly that the question remains open even for height 1 (weakly malnormal $H$).
Verified sources (each checked against Crossref or the arXiv API):
R. Gitik, M. Mitra, E. Rips, M. Sageev, "Widths of Subgroups", Trans. Amer. Math. Soc. 350(1) (1998), 321–329. DOI: 10.1090/S0002-9947-98-01792-9. (Verified via Crossref.) Introduces width/height of subgroups and proves that a quasiconvex subgroup of a word-hyperbolic group has finite width (hence finite height). This is the "converse" mentioned in Bestvina's list: quasiconvexity implies the conjugate-intersection hypothesis of Q 1.8.
I. Kapovich, H. Short, "Greenberg's Theorem for Quasiconvex Subgroups of Word Hyperbolic Groups", Canad. J. Math. 48(6) (1996), 1224–1244. DOI: 10.4153/CJM-1996-065-6. (Verified via Crossref.) Hyperbolic-group analogue of Greenberg's theorem: a quasiconvex subgroup $H$ has finite index in its commensurator (virtual normalizer) $\mathrm{Comm}_G(H)$; in particular $[N_G(H):H]<\infty$. Thus all three of Swarup's hypotheses ($H$ finitely presented; $[N_G(H):H]<\infty$; finite height) are necessary conditions for quasiconvexity; Q 1.8 asks whether they are jointly sufficient.
M. Mitra, "Height in splittings of hyperbolic groups", Proc. Indian Acad. Sci. (Math. Sci.) 114(1) (2004), 39–54. DOI: 10.1007/BF02829670; arXiv:math/0403125. (Verified via Crossref and arXiv API.) Answers Swarup's question affirmatively in the split case: if $H$ is a hyperbolic subgroup of a hyperbolic group $G$, the intersection of any $n$ essentially distinct conjugates of $H$ is finite, $G$ splits over $H$ with hyperbolic vertex and edge groups, and the two inclusions of $H$ are quasi-isometric embeddings, then $H$ is quasiconvex in $G$. The paper also formulates a chain of successively stronger properties of a non-quasiconvex subgroup (infinite height, strictly infinite height, a "strong" version with intersections along powers of one element) and proves implications between them, so that a negative answer to Swarup's question would yield subgroups with these exotic intersection patterns. Caveat: the theorem presupposes $H$ hyperbolic and quasi-isometrically embedded in the vertex groups — the general question, where $H$ is only finitely presented (and a priori possibly non-hyperbolic and distorted), is untouched.
A. Pal, "Height in splittings of relatively hyperbolic groups", Geom. Dedicata 213 (2021), 121–135. DOI: 10.1007/s10711-020-00571-1. (Verified via Crossref.) Extends Mitra's split-case theorem to relatively hyperbolic groups.
C. Abbott, E. Martínez-Pedroza, "The quasi-isometry invariance of the Coset Intersection Complex", Algebr. Geom. Topol. 26 (2026), 659–698. DOI: 10.2140/agt.2026.26.659; arXiv:2404.16628. (Verified via arXiv API, including journal ref.) Builds a simplicial complex encoding finite height / finite width / almost malnormality and proves these properties are quasi-isometry invariants of the pair $(G,H)$; the authors explicitly frame parts of their main theorem as "evidence of a positive answer to Swarup's question" — i.e., they treat the question as open.
R. Halder, P. Sardar, "Embeddings of trees of hyperbolic metric spaces and Cannon–Thurston maps", arXiv:2511.12883 (v1 Nov 2025, v3 Feb 2026). (Verified via arXiv API.) States plainly: "even if $H$ is of height 1 in $G$, i.e. $H$ is weakly malnormal in $G$, Swarup's question remains open. It is known only in certain special cases." Proves existence of Cannon–Thurston maps for certain amalgams $K_1 *_H K_2 \to G_1 *_H G_2$ (a weakening of the conclusion of quasiconvexity), continuing the Mitra/Pal line.
M. Mitra, "Coarse extrinsic geometry: a survey", in The Epstein Birthday Schrift, Geom. Topol. Monogr. 1 (1998), 341–364. DOI: 10.2140/gtm.1998.1.341; arXiv:math/9810203. (Verified via Crossref.) Survey that records Swarup's question and the state of knowledge circa 1998.
I. Kapovich, "A non-quasiconvex subgroup of a hyperbolic group with an exotic limit set", New York J. Math. 1 (1995). (Verified at https://nyjm.albany.edu/j/1995/1-12p.pdf.) Records the related theorem attributed to Swarup: a finitely presented one-ended subgroup of a word-hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer (commensurator). This is the strongest known "purely algebraic" criterion for quasiconvexity and is the backdrop of Q 1.8: Swarup asks whether the commensurator hypothesis can be weakened to the normalizer hypothesis at the price of adding finite height. (The underlying Swarup preprint appears never to have been formally published; the statement survives through this citation.)
Work done
- Read the dataset item; identified the source as Bestvina's problem list Q 1.8 and confirmed the original wording (including Gitik's remark that the malnormal case is open) against the indexed text of the author's PDF.
- Verified every citation above against Crossref (DOIs 10.1090/S0002-9947-98-01792-9, 10.4153/CJM-1996-065-6, 10.1007/BF02829670, 10.1007/s10711-020-00571-1, 10.2140/gtm.1998.1.341) or the arXiv API (math/0403125, 2404.16628 with journal ref AGT 26 (2026) 659–698, 2511.12883), plus the NYJM page for Kapovich 1995.
- Established current status from the two most recent sources (Abbott–Martínez-Pedroza 2026, Halder–Sardar 2025/2026), both of which treat the question as open.
- Analyzed the logical structure of the hypotheses (see Result).
Result
The question is open; I cannot solve it, but the literature plus elementary reasoning gives a clean triage.
1. The hypotheses are exactly the known necessary conditions. For $H$ a subgroup of a hyperbolic group $G$: quasiconvex $\Rightarrow$ $H$ finitely presented (standard), quasiconvex $\Rightarrow$ finite height/width (GMRS 1998), and quasiconvex $\Rightarrow$ $[\mathrm{Comm}_G(H):H]<\infty$, hence $[N_G(H):H]<\infty$ (Kapovich–Short 1996). Swarup's question is precisely whether this conjunction of necessary conditions is sufficient. Degenerate cases are trivial: if $[G:H]<\infty$ or $H$ is finite, $H$ is quasiconvex; so the content is for infinite-index infinite $H$.
2. Reductions and equivalences. Because $[N_G(H):H]<\infty$, the hypothesis "$n$ distinct conjugates have finite intersection" is the same as "height$(H)\le n-1$" up to the bounded factor $[N_G(H):H]$; the $n=2$ case is almost malnormality (malnormality in the torsion-free case). Gitik's remark in the list, and Halder–Sardar twenty years later, both record that even the (weakly) malnormal case is open.
3. Where the difficulty lies. Swarup's own virtual-normalizer criterion (item 8 above) shows that for finitely presented one-ended $H$, finite index in the commensurator suffices for quasiconvexity. The gap in Q 1.8 is twofold: (a) the normalizer can be much smaller than the commensurator — finite height is meant to compensate by bounding how many conjugates can share an infinite intersection (each commensurator coset yields such a conjugate), but no proof currently upgrades "height $\le n-1$ + $[N_G(H):H]<\infty$" to "$[\mathrm{Comm}_G(H):H]<\infty$"; (b) $H$ is only assumed finitely presented, not one-ended or even hyperbolic, so the one-ended criterion does not apply directly, and the ends/many-ended case requires separate arguments (splittings of $H$ over finite groups, where Mitra's theorem is exactly the relevant tool — this is presumably why the split case fell first).
4. Known distortion mechanisms do not give counterexamples. The classical non-quasiconvex finitely presented subgroups of hyperbolic groups — Rips-construction kernels (normal of infinite index, so $[N_G(H):H]=\infty$) and Brady-type non-hyperbolic f.p. subgroups — violate at least one of Swarup's hypotheses in all known instances. Whether this is forced (positive answer) or whether more exotic constructions (finitely presented, distorted, yet finite-height and almost normal) exist (negative answer) is the unresolved core.
5. Partial positive results. Mitra (2004) settled the split case ($G$ splits over $H$, inclusions quasi-isometric), Pal (2021) its relatively hyperbolic version; Halder–Sardar (2025) obtain Cannon–Thurston maps (a boundary-level shadow of quasiconvexity) for further amalgam configurations; Abbott–Martínez-Pedroza (2026) show the hypotheses of the question are robust under quasi-isometry, which they interpret as evidence for a positive answer.
What remains
- The full question is open, and by Gitik's remark (still endorsed in the 2025 literature) even the case $n=2$, i.e. $H$ (almost) malnormal, finitely presented, with $[N_G(H):H]<\infty$, is undecided.
- Natural next steps: (i) settle whether "finite height + $[N_G(H):H]<\infty$" implies $[\mathrm{Comm}_G(H):H]<\infty$, which by Swarup's criterion would answer the one-ended case affirmatively; (ii) handle many-ended $H$ via its JSJ/grushko splitting over finite groups combined with Mitra-type combination arguments; (iii) on the negative side, attempt Rips/Brady-style constructions with controlled height — Mitra's 2004 paper shows any counterexample must have (strictly) infinite height analogues among its conjugate-intersection patterns, which constrains the geometry such a construction must exhibit.
- A confirmed answer either way would close one of the last open items of the "algebraic characterization of quasiconvexity" program from the 1990s (GMRS/Kapovich–Short/Swarup).