research/AMR-010-0108.md

---
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):

1. **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.

2. **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**.

3. **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.

4. **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.

5. **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.

6. **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.

7. **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.

8. **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).