research/AMR-010-0101.md

metadata

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).