research/AMR-010-0116.md

metadata

id: AMR-010-0116
classification: OPEN-TRIAGE
wording_corrected: 'no'

AMR-010-0116 — Rank of direct powers of a hyperbolic group (Dani Wise's conjecture)

Problem (corrected statement if needed)

The dataset transcription is accurate. The original wording, from M. Bestvina, Questions in Geometric Group Theory (updated July 2004), Question 1.16 (author PDF, verified against the fetched source):

(Dani Wise) Let $G^n$ denote the Cartesian product of $n$ copies of the group $G$, and let $\operatorname{rank}(G^n)$ be the smallest number of generators of $G^n$. Conjecture. If $G$ is word-hyperbolic then $\displaystyle\lim_{n\to\infty}\operatorname{rank}(G^n)=\infty$.

The notes in the source add: the conjecture is true for finite nontrivial $G$ by a pigeon-hole argument; it is true whenever it holds for some quotient of $G$ (in particular when $G$ has a proper finite-index subgroup); and it fails if there is an epimorphism $G \twoheadrightarrow G\times G$, with Wise having an example of a 2-generator infinitely presented $C'(1/6)$ small cancellation group witnessing this. The companion question Q 1.17 asks for "nice" (CAT(0), automatic, ...) groups where the conjecture fails.

The question originates in Wise's own paper: D. T. Wise, The rank of a direct power of a small-cancellation group, Geom. Dedicata 94 (2002), 215–223, doi:10.1023/A:1020968914280 (existence and abstract verified via Springer and the Weizmann Institute publication record).

Status / Literature

Open, as far as I could verify (August 2026). I found no paper resolving the conjecture in either direction; recent surveys and papers that cite Wise's conjecture (e.g. Coulon–Fournier-Facio 2023, below) still treat growth sequences of infinite groups as "mysterious". The following literature is verified (each item was checked against the arXiv API, the publisher page, or the reference list of a verified paper):

• D. T. Wise (2002), The rank of a direct power of a small-cancellation group, Geom. Dedicata 94, 215–223. Verified abstract: he constructs (i) a finitely generated $C'(1/6)$ group $G_\infty$ with $\operatorname{rank}(G_\infty^n)=2$ for all $n$ — so the conjecture fails badly for infinitely presented small cancellation groups; (ii) for each fixed $n$ a finitely presented $C'(1/6)$ group $G_n$ with $\operatorname{rank}(G_n^n)=2$; (iii) a finitely generated $C'(1/6)$ group $D$ with an epimorphism $D\twoheadrightarrow D\times D$; and (iv) for each $m$ a residually finite $C'(1/6)$ group with no proper subgroups of index $\le m$. He explicitly conjectures the positive statement for word-hyperbolic groups.
• J. Wiegold & J. S. Wilson (1978), Growth sequences of finitely generated groups, Arch. Math. (Basel) 30, 337–343 (MR 503347; verified as cited in Coulon–Fournier-Facio's reference list). They prove: for a finitely generated infinite simple group $\Gamma$, $d(\Gamma^p)\le d(\Gamma)+1$ for all $p\ge 1$ — infinite simple groups have essentially bounded growth sequences, and no f.g. infinite simple group with non-constant growth sequence is known (Wiegold–Wilson call this "irreducibly difficult"; see also Wiegold, Is the direct square of every 2-generator simple group 2-generator?, Publ. Math. Debrecen 35 (1988), 207–209).
• A. Yu. Olshanskii (1995), SQ-universality of hyperbolic groups, Mat. Sb. 186 (verified indirectly: invoked as [Ol'95] in Coulon–Fournier-Facio for the SQ-universality of torsion-free non-elementary hyperbolic groups).
• R. Coulon & F. Fournier-Facio (2023), Infinite simple characteristic quotients, arXiv:2312.11684 (verified via the arXiv API and the fetched paper). Theorem 1.5/4.1: every torsion-free non-elementary hyperbolic group $\Gamma$ admits infinite, simple, characteristic quotients $\Gamma/N$ — all not finitely presentable — containing any prescribed countable group. Combined with Wiegold–Wilson, these simple quotients $S$ satisfy $d(S^p)\le d(S)+1\le d(\Gamma)+1$ for all $p$.
• The finite case (nontrivial finite $F$ has $d(F^n)\to\infty$) is classical growth-sequence theory initiated by Wiegold; Bestvina's notes record the elementary pigeon-hole argument.

Work done

I verified the source wording (exact match; no correction needed), established the above verified literature base, and carried out the following rigorous reductions and observations (elementary but, as far as I can tell, the correct state of knowledge):

Proposition (reduction of the conjecture). Let $G$ be a finitely generated group. Then $\operatorname{rank}(G^n)\to\infty$ in each of the following cases:

1. $b_1(G)=\operatorname{rank}{\mathbb Z}(G{ab})\ge 1$: then $(G^n){ab}=G{ab}^n$ surjects $\mathbb Z^{,n,b_1(G)}$, so $\operatorname{rank}(G^n)\ge n,b_1(G)\to\infty$.
2. $G$ has some nontrivial finite quotient $F$: then $\operatorname{rank}(G^n)\ge \operatorname{rank}(F^n)\to\infty$ by the finite case. In particular this holds if $G$ has a proper finite-index subgroup (take its core).

Proof. (1) is immediate since $\operatorname{rank}$ does not increase under quotients and $\operatorname{rank}(\mathbb Z^{m})=m$. For (2), $\operatorname{rank}(G^n)\ge \operatorname{rank}(Q^n)$ for every quotient $Q$ of $G$. $\square$

Corollary 1. A counterexample to Wise's conjecture must be an infinite word-hyperbolic group that is perfect ($b_1=0$) and has no nontrivial finite quotients at all. In particular it must fail to be residually finite: since residual finiteness of an infinite group produces arbitrarily large finite quotients, a positive answer to Bestvina's Q 1.15 (every hyperbolic group is residually finite — itself famously open) would imply Wise's conjecture. So Q 1.16 is a strict weakening of Q 1.15, and any counterexample to Q 1.16 is also a counterexample to Q 1.15.

Corollary 2. If $G\twoheadrightarrow G\times G$, then iterating gives $G\twoheadrightarrow G^{2^k}$, so $\operatorname{rank}(G^{2^k})\le\operatorname{rank}(G)$ and the conjecture fails for $G$. Wise's 2002 examples show such epimorphisms exist in the $C'(1/6)$ class when finite presentability is dropped. No hyperbolic (finitely presented) group $G$ with an epimorphism onto $G\times G$ is known, and no obvious invariant rules one out.

Observation 3 (the simple-group route is closed inside hyperbolic groups). By Olshanskii's SQ-universality, every non-elementary hyperbolic group has every countable group embedded in some quotient. A simple group $S$ has only the quotients $S$ and $1$; if $S$ were a non-elementary hyperbolic group, every countable group would embed in $S$ itself — impossible, since the finitely generated $S$ has only countably many finitely generated subgroups while there are uncountably many isomorphism classes of finitely generated groups. Hence no infinite simple hyperbolic group exists, and the Wiegold–Wilson mechanism ($d(\Gamma^p)\le d(\Gamma)+1$ for infinite simple $\Gamma$) cannot produce a hyperbolic counterexample. Conversely, Coulon–Fournier-Facio show that every torsion-free non-elementary hyperbolic group does have (non-finitely-presentable) infinite simple quotients, whose growth sequences are bounded by Wiegold–Wilson. So quotient-based lower bounds on $\operatorname{rank}(G^n)$ coming from simple quotients cannot prove the conjecture; only finite quotients or the abelianization can, and any proof must use finite presentability of $G$ essentially (hyperbolicity of quotients alone is insufficient, since non-finitely-presentable $C'(1/6)$ counterexamples exist).

Observation 4 (homological lower bounds fail). The only Betti number giving a usable bound is $b_1$: $\operatorname{rank}(H)\ge b_1(H)$, and by Künneth $b_1(G^n)=n,b_1(G)$ — this is exactly case (1). Higher homology gives nothing: there is no inequality $\operatorname{rank}(H)\ge b_2(H)-b_1(H)$ (e.g. $H=\mathbb Z\wr\mathbb Z$ is 2-generated with $H_2(H)$ free abelian of infinite rank, by the standard exterior-square computation of $H_2$ of a wreath product), and the naive Euler-characteristic bound "$\chi(H)\ge 1-\operatorname{rank}(H)$" fails already for $H=F_4^3$ ($\chi=(-3)^3=-27$ but $\operatorname{rank}=12<28=1-\chi$), since it requires cohomological dimension $\le 2$. Likewise $L^2$-Betti numbers of $G^n$ vanish for infinite $G$ (Cheeger–Gromov), so $L^2$ methods give no rank bound.

Result

Wise's conjecture (Bestvina Q 1.16) remains open. I did not solve it. What is established here:

• Verified the original statement and the absence of a published solution; the only published partial results are Wise's 2002 small-cancellation counterexamples outside the finitely presented/hyperbolic world, and the classical finite-group growth-sequence theory.
• Sharp reduction (Proposition + Corollary 1): the conjecture holds unless $G$ is infinite, perfect, and has no nontrivial finite quotients; hence it is implied by residual finiteness of hyperbolic groups (Q 1.15), and a counterexample would simultaneously refute Q 1.15.
• Structural observations: no infinite simple hyperbolic group exists (Observation 3), so the known bounded-growth mechanism for infinite simple groups cannot realize a hyperbolic counterexample; but every torsion-free non-elementary hyperbolic group has non-finitely-presentable simple quotients with bounded growth sequences (Coulon–Fournier-Facio + Wiegold–Wilson), so any proof must exploit finite presentability in an essential way. Homological/$L^2$ invariants cannot detect rank growth beyond $b_1$ (Observation 4).

What remains

• Decide the conjecture in the residual case: $G$ infinite hyperbolic, $b_1(G)=0$, with no nontrivial finite quotients. This is entangled with the residual finiteness problem (Q 1.15): proving all hyperbolic groups residually finite settles Wise's conjecture affirmatively; constructing a hyperbolic group with an epimorphism $G\twoheadrightarrow G^2$ (or even with $\operatorname{rank}(G^n)$ bounded) would refute both.
• Already the case $n=2$ is open in general: is $\operatorname{rank}(G^2)> \operatorname{rank}(G)$ (or even $\ge \operatorname{rank}(G)+1$) for every non-elementary hyperbolic $G$ with $b_1(G)=0$?
• No growth rate is known in the cases where the conjecture holds only via finite quotients: lower bounds on $\operatorname{rank}(G^n)$ in terms of the finite-quotient growth of $G$ would be quantitative strengthenings (for finite simple $S$ one knows $\operatorname{rank}(S^n)=\Theta(\log n)$-type behavior from Wiegold's theory).
• Honesty note: the non-existence of a published solution is asserted on the basis of targeted searches (arXiv API, web) rather than exhaustive review; if a resolution appeared very recently or in an obscure venue, I did not find it. The citation [Bri22] in Coulon–Fournier-Facio's introduction (approaches "of a different flavor" to growth sequences of infinite groups) was not independently identified or verified.