research/AMR-010-0207.md

metadata

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.