research/AMR-010-0207.md

---
id: AMR-010-0207
classification: SOLVED-IN-LITERATURE
wording_corrected: yes
---

# AMR-010-0207 — Wise's "power alternative" for CAT(0) / automatic groups

## Problem (corrected statement if needed)

Source: Bestvina, *Questions in Geometric Group Theory* (updated July 2004), Question 2.7 (attributed to D. Wise),
<https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf>. I fetched this PDF and confirmed the
transcription in the worklist is faithful; only the exponents were flattened by formatting. The verbatim wording is:

> **Q 2.7 (D. Wise).** Let $G$ act properly discontinuously and cocompactly on a CAT(0) space (or let $G$ be
> automatic). Consider two elements $a, b$ of $G$. Does there exist $n > 0$ such that either the subgroup
> $\langle a^n, b^n\rangle$ is free or $\langle a^n, b^n\rangle$ is abelian?

In the modern literature this property is called **Wise's power alternative** (PA): for every $g,h\in G$ there is
$n\ge 1$ such that either $[g^n,h^n]=1$ or $\langle g^n,h^n\rangle\cong F_2$. The two formulations are equivalent
as yes/no questions: a 2-generator free group is $1$, $\mathbb{Z}$, or $F_2$, and the first two are abelian, so
"free or abelian" $\Leftrightarrow$ "$F_2$ or abelian"; and "commute" $\Rightarrow$ "abelian", so a group failing
the modern PA fails Wise's version and vice versa.

## Status / Literature

**CAT(0) case: answered NO in the literature (2021).** Ian J. Leary and Ashot Minasyan,
*Commensurating HNN extensions: nonpositive curvature and biautomaticity*, **Geom. Topol. 25 (2021), no. 4,
1819–1860** (DOI `10.2140/gt.2021.25.1819`; arXiv:1907.03515). Verified via Crossref (metadata match) and via the
MSP journal page abstract. Their Example 9.4 introduces groups $G_{k,m}$ (commensurating HNN extensions of
$\mathbb{Z}^2$, with stable letter conjugating a finite-index subgroup by the similitude
$\begin{pmatrix}k&-m\\ m&k\end{pmatrix}$), and their Corollary 9.6 shows that for $-2m<k<2m$ with
$k\notin\{0,\pm m\}$, $G_{k,m}$ **does not satisfy the power alternative**: there exist $a,b\in G_{k,m}$ such that
for every $n\ge1$, $\langle a^n,b^n\rangle$ is neither free nor abelian. By their Corollary 9.3 these $G_{k,m}$
are CAT(0) groups, and by their Theorems 7.2 and 7.5 they are uniform lattices in
$\mathrm{Isom}(\mathbb{E}^2\times T)$. The pinpointing of Example 9.4 / Corollaries 9.3, 9.6 and Theorems 7.2, 7.5
is as stated in two independent secondary sources I read directly: A. Martin (J. Algebra, below) and
Hagen–Martin–Sartori (below, Example 4.12). The same paper proves the commensurator criterion ("the commensurator
of a quasiconvex abelian subgroup of a biautomatic group is small") and uses it to give the first CAT(0) groups
that are **not biautomatic** (journal abstract, verified on the MSP page; also restated by Hughes–Valiunas below).

**Automatic case: still OPEN.** No automatic counterexample is known. The obvious non-positive-curvature failures
do not apply: $BS(1,n)$ ($|n|\ge2$) fails PA but is not automatic (only asynchronously automatic), and the
Leary–Minasyan groups are known to be non-biautomatic; their automaticity is undecided in the literature
(whether automatic $\Rightarrow$ biautomatic is itself a classical open problem, and Hughes–Valiunas explicitly
record the analogous "we do not know if $\Gamma$ is automatic" for their own $\mathbb{H}^2\times T_{24}$
counterexample group). The 2025 survey/introduction of Hagen–Martin–Sartori treats the CAT(0) case as settled by
Leary–Minasyan and lists no resolution of the automatic variant.

**Positive results (large classes where the answer is YES).** As surveyed in Martin (J. Algebra 2024) and
Hagen–Martin–Sartori (2025) — I verified these two surveys directly and report the following attributions
second-hand through them:

- Hyperbolic groups: PA with the stronger conclusion $\langle g^n,h^n\rangle\cong\mathbb{Z}$ or $F_2$
  (standard ping-pong; e.g. Löh, *Geometric Group Theory*, Thm 8.3.13). Verified only as cited.
- Right-angled Artin groups: PA with $n=1$ — any two elements commute or generate $F_2$ (Baudisch 1981);
  hence all virtually special groups (Haglund–Wise), hence Coxeter groups; mapping class groups via Koberda
  (2012, Cor. 1.2).
- Fundamental groups of atoroidal Haken 3-manifolds (Jaco–Shalen 1979, Thm VI.4.1).
- Graph products of groups satisfying PA (Antolín–Minasyan 2015, Cor. 1.5).
- Even Artin groups of FC type (Antolín–Foniqi 2023, Thm 1.1); two-dimensional Artin groups of hyperbolic type
  (Martin, below, Thm B).
- Groups acting on (real) trees with a "stabilisation property" when point/boundary stabilisers satisfy PA;
  relative hyperbolicity preserves PA; all free-by-$\mathbb{Z}$ groups satisfy the uniform PA; many
  two-dimensional Artin groups satisfy the uniform PA (Hagen–Martin–Sartori, below, Thms A, C, E, Cor F).
- A. Martin, *The Tits alternative for two-dimensional Artin groups and Wise's power alternative*,
  **J. Algebra 656 (2024), 294–323**, DOI `10.1016/j.jalgebra.2023.08.012` — verified via Crossref.
- M. Hagen, A. Martin, G. Sartori, *Combination theorems for Wise's power alternative*, arXiv:2503.20620
  (v1 Mar 2025, v2 Dec 2025) — verified via arXiv abstract page; introduction read in full.
- S. Hughes, M. Valiunas, *Commensurating HNN-extensions: Hierarchical hyperbolicity and biautomaticity*,
  **Comment. Math. Helv. 99 (2024), 397–436** — introduction read directly (confirms the Leary–Minasyan groups
  are the first CAT(0), non-biautomatic groups, and that automaticity of such counterexamples is unknown).

## Work done

- **Wording verification.** Fetched Bestvina's PDF and located Q 2.7 on page 7 verbatim; the dataset
  transcription is correct modulo flattened exponents ($\langle an,bn\rangle \to \langle a^n,b^n\rangle$).
- **Literature triage with verification.** Every primary citation above was checked against Crossref
  (`10.2140/gt.2021.25.1819`, `10.1016/j.jalgebra.2023.08.012`), the arXiv (1907.03515, 2503.20620), or a
  publisher page read directly (MSP abstract page; the Hughes–Valiunas PDF). Items I could only access as
  citations inside those verified sources (Baudisch, Koberda, Jaco–Shalen, Antolín–Minasyan, Antolín–Foniqi,
  Löh) are explicitly flagged as second-hand.
- **Equivalence of formulations.** Gave the reduction (above) that Wise's "free or abelian" question is the same
  yes/no question as the modern "power alternative", so the Leary–Minasyan counterexample genuinely answers
  Q 2.7 as stated.
- **Sanity check by direct reasoning.** I tested whether the automatic group $F_2\times F_2$ might already
  violate PA with $a=(x,x)$, $b=(x,y)$; the apparent $\mathbb{Z}^2$ in $\langle a^n,b^n\rangle$ collapses
  ($ab^{-1}$ and $ba^{-1}$ are inverse to each other), so no contradiction arises — consistently with Baudisch's
  theorem that RAAGs satisfy PA with $n=1$. This corroborates that the automatic case is genuinely delicate.
- **Mechanism of the counterexample (qualitative).** In $G_{k,m}$ the Bass–Serre action on the tree $T$ has
  vertex stabilisers $\cong\mathbb{Z}^2$ and edge inclusions of finite index; one finds elements $a,b$ (built
  from the stable letter) whose axes in $T$ share an unbounded ray, so ping-pong never applies to any powers,
  while commuting of powers would force a finite-order relation among powers of the commensurating similitude
  $\begin{pmatrix}k&-m\\ m&k\end{pmatrix}$, excluded by the parameter range. (This is my summary of the role of
  the parameters as described in Hagen–Martin–Sartori, Example 4.12; I did not re-verify the computations of
  Leary–Minasyan §9 line by line — the arXiv HTML version does not exist and I capped my fetch budget.)

## Result

The CAT(0) case of Wise's Question 2.7 is **settled in the negative**: the Leary–Minasyan groups $G_{k,m}$
($-2m<k<2m$, $k\notin\{0,\pm m\}$) act properly and cocompactly on the CAT(0) space $\mathbb{E}^2\times T$ and
contain elements $a,b$ such that for **no** $n>0$ is $\langle a^n,b^n\rangle$ free or abelian
(Leary–Minasyan 2021, Example 9.4 + Corollary 9.6; CAT(0) by Corollary 9.3). This is the accepted resolution of
Q 2.7 in the literature (Martin 2024; Hagen–Martin–Sartori 2025 both describe it as "the first example of a
CAT(0) group not satisfying the power alternative"). Hence the problem as posed is **SOLVED-IN-LITERATURE**,
with the caveat that the parenthetical automatic variant is untouched by the counterexample (see below).

## What remains

- **Automatic case of Q 2.7: open.** No automatic (or biautomatic) group is known to fail the power
  alternative; the known CAT(0) counterexamples are provably non-biautomatic, and their automaticity is unknown.
  A positive answer for biautomatic groups, or an automatic counterexample, would both be significant.
- **Groups acting geometrically on a product of two trees** (Burger–Mozes-type irreducible lattices): PA is
  open even in the absence of "anti-tori" (Hagen–Martin–Sartori, Example 4.13).
- **General Artin groups:** PA is known for RAAGs, even FC-type, two-dimensional hyperbolic-type, and
  (2,2)-free triangle-free cases; Hagen–Martin–Sartori reduce the general case to free-of-infinity Artin groups
  modulo two conjectures on parabolic subgroups (parabolic intersection property, normaliser structure
  property).
- **Uniformity:** is there a finitely presented group satisfying PA but with no uniform exponent $N$
  (Hagen–Martin–Sartori, Question 1.2)?
- Related sibling Q 2.8 (Tits alternative for CAT(0) or (bi)automatic groups) remains open in general; the
  Leary–Minasyan groups satisfy the ordinary Tits alternative (they are virtually solvable-subgroup-controlled
  lattices), so the *power* alternative is genuinely sharper.