research/AMR-010-0101.md

---
id: AMR-010-0101
classification: SOLVED-IN-LITERATURE
wording_corrected: no
---

# AMR-010-0101 — Bestvina Q 1.1: finite K(G,1), no Baumslag–Solitar subgroups ⟹ hyperbolic?

## Problem (corrected statement if needed)

From M. Bestvina, *Questions in Geometric Group Theory* (2004), Question 1.1
(https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf):

> Suppose $G$ admits a finite $K(G,1)$. If $G$ does not contain any
> Baumslag–Solitar subgroups $BS(m,n)$, is $G$ necessarily hyperbolic?
> If $G$ embeds in a hyperbolic group, is it hyperbolic?

The transcription in the source file is accurate; no correction was needed.
The condition "no $BS(m,n)$" is a *necessary* condition for hyperbolicity
(hyperbolic groups contain no $BS(m,n)$: $BS(m,\pm m)$ contains $\mathbb{Z}^2$
up to finite index, and $BS(m,n)$ with $|m|\neq|n|$ is solvable but not
virtually cyclic); the question asks whether it is sufficient, given a finite
$K(G,1)$.

## Status / Literature

**Both questions are answered negatively** by:

- G. Italiano, B. Martelli, M. Migliorini, *Hyperbolic 5-manifolds that fiber
  over $S^1$*, Invent. Math. **231** (2023), 1–38; arXiv:2105.14795
  (first posted May 2021). Publication venue confirmed on B. Martelli's
  publication list (open access: doi 10.1007/s00222-022-01141-w).

The paper's Corollary 2 states: *there is a hyperbolic group $G$ containing a
subgroup $H$ of finite type that is not hyperbolic*, where "finite type" is
defined exactly as "fundamental group of a finite aspherical cell complex",
i.e. $H$ admits a finite $K(H,1)$ (in fact $\mathrm{cd}(H)=4$,
$\mathrm{cd}(G)=5$). This answers the **second** question. Corollary 3 states:
*there is a finite type group $H$ that is not hyperbolic and does not contain
any Baumslag–Solitar subgroup $BS(m,n)$* — immediate since $H$ lies inside a
hyperbolic group. This answers the **first** question. The authors explicitly
note that the pair $H<G$ was raised as Bestvina's Question 1.1 (and also as
Brady's Question 7.2, Bridson's Question 4.1, and by Jankiewicz–Norin–Wise),
and Corollary 3 as a question of Bestvina, Bridson (Q 2.22), and
Druțu–Kapovich (Problem 11.129).

Prior milestone: N. Brady, *Branched coverings of cubical complexes and
subgroups of hyperbolic groups*, J. London Math. Soc. **60** (1999), 461–480,
built a finitely presented non-hyperbolic subgroup of a hyperbolic group, but
his group is not of type $FP_3$, hence has no finite $K(G,1)$ — so the
finite-$K(G,1)$ version remained open until 2021.

Known *positive* special cases (both cited in IMM): every finite type subgroup
of a hyperbolic group of cohomological dimension 2 is hyperbolic (Gersten);
and within the classes of free-by-cyclic groups (Brinkmann) and ascending HNN
extensions of free groups, "hyperbolic ⟺ contains no Baumslag–Solitar
subgroup" holds.

## Work done

I verified the statement against Bestvina's list, located the resolving paper
via arXiv search, and read the construction and proofs in the published
argument (arXiv:2105.14795v4 full text). Sketch of the counterexample:

1. **A fibering hyperbolic 5-manifold.** Using Bestvina–Brady PL Morse theory
   and the states/moves game of Jankiewicz–Norin–Wise on the cubulation dual
   to a tessellation by the right-angled hyperbolic 5-polytope $P^5$, they
   produce a cusped finite-volume hyperbolic 5-manifold $M^5$ (commensurable
   with the Ratcliffe–Tschantz manifold, the smallest known hyperbolic
   5-manifold) that fibers over $S^1$. The fiber $F^4$ is aspherical with
   $\chi(F)=1$.
2. **Killing the cusps by filling.** Truncate $M^5$ to a compact $\bar M^5$
   whose boundary consists of flat 4-tori; isotope the fibration so it
   restricts on each boundary 4-torus to a fibration by geodesic 3-tori. Pass
   to a finite cover so all boundary 4-tori have systole $>2\pi$, then shrink
   each 3-torus fiber to a point. The resulting space $\hat M^5$ is an
   aspherical pseudo-manifold, and by the Fujiwara–Manning filling theorem
   (Thm 2.7 of their ref. [15]) it carries a locally CAT($-\kappa$) metric, so
   $G=\pi_1(\hat M^5)$ is hyperbolic and torsion-free.
3. **The subgroup $H$.** The fibration descends to $\hat M^5\to S^1$ whose
   fiber $\hat F^4$ is $\bar F^4$ with each boundary 3-torus coned to a point.
   $\hat F^4$ is aspherical (its product with $\mathbb{R}$ covers $\hat M^5$),
   so $H=\pi_1(\hat F^4)=\ker(G\to\mathbb{Z})$ has a finite 4-dimensional
   $K(H,1)$ — it is of finite type, and $H<G$.
4. **$H$ is not hyperbolic.** $H^4(H)=\mathbb{Z}$ (pseudo-manifold top class)
   and $\mathrm{Out}(H)$ is infinite: powers of the monodromy are nontrivial
   in $\mathrm{Out}(H)$, using that centralizers of nontrivial elements in the
   hyperbolic group $G$ are cyclic. If $H$ were hyperbolic, Rips theory
   (Bestvina–Feighn, *Stable actions of groups on real trees*, Cor. 1.3) would
   split $H$ over a cyclic subgroup; but a Mayer–Vietoris computation shows
   this is impossible, because the vertex groups have infinite index in $H$,
   hence are $\pi_1$ of noncompact aspherical 4-dimensional covers and have
   $H^4=0$, contradicting $H^4(H)=\mathbb{Z}$.
5. **No Baumslag–Solitar subgroups.** $H$ embeds in the hyperbolic group $G$,
   and hyperbolic groups contain no $BS(m,n)$; hence $H$ is a finite type,
   non-hyperbolic group with no Baumslag–Solitar subgroups.

Note the role of the coning in step 2–3: the raw fiber group $\pi_1(\bar F^4)$
*does* contain $\mathbb{Z}^4$ cusp subgroups (hence $BS(1,1)=\mathbb{Z}^2$);
only after the $2\pi$-filling does one get a group inside a hyperbolic group.

## Result

Both parts of Bestvina's Question 1.1 have answer **no**:
- There exists a group $H$ with a finite $K(H,1)$ (indeed a finite aspherical
  4-dimensional complex) that contains no Baumslag–Solitar subgroup $BS(m,n)$
  yet is not hyperbolic.
- The same $H$ embeds in a hyperbolic group $G$ ($\pi_1$ of a filled,
  CAT($-\kappa$) 5-dimensional pseudo-manifold, $\mathrm{cd}(G)=5$).

Reference: Italiano–Martelli–Migliorini, Invent. Math. 231 (2023), 1–38,
arXiv:2105.14795, Corollaries 2 and 3.

## What remains

The original questions are settled, but natural strengthenings remain open
(raised in IMM §4 and elsewhere):
- Are there *closed* (compact) hyperbolic manifolds of dimension $\ge 5$ that
  fiber over $S^1$? All known examples in dimension 5 are cusped; a closed
  fibering example would give a counterexample $H$ that is a closed aspherical
  manifold group.
- The minimal (cohomological) dimension of such a counterexample: the IMM
  example has $\mathrm{cd}(H)=4$ inside $\mathrm{cd}(G)=5$, while finite type
  subgroups of hyperbolic groups with $\mathrm{cd}=2$ are always hyperbolic
  (Gersten). The intermediate dimensions and the question whether $H$ can be
  taken to be a *manifold* group remain of interest (cf. subsequent work of
  the same authors on algebraic fibering up to dimension 8).