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Combination theorems for Wise's power alternative

Mark Hagen, Alexandre Martin, Giovanni Sartori

TL;DR

This work develops a unified framework of Wise's power alternative for groups acting on tree-like objects, showing WPA can be transferred from stabilisers to the whole group under a stabilisation property. It then applies this to Artin groups via visual splittings, reducing WPA to conjectural parabolic properties and proving a uniform WPA for a substantial class of (2,2)-free triangle-free Artin groups, with explicit uniform exponents and consequences such as uniform exponential growth. A parallel theory for relatively hyperbolic groups shows WPA (and uniform WPA under torsion bounds) passes from peripheral subgroups to the ambient group, with implications for acylindrical hyperbolicity and HHGs. The paper also develops a detailed set of tools (ultimate translation length, law-based variants, and mapping-tori arguments) to carry WPA through combination constructions, yielding broad new instances and several natural questions about the scope and limits of uniformity and hierarchical hyperbolicity.

Abstract

We show that Wise's power alternative is stable under certain group constructions, use this to prove the power alternative for new classes of groups, and recover known results from a unified perspective. For groups acting on trees, we introduce a dynamical condition that allows us to deduce the power alternative for the group from the power alternative for its stabilisers of points. As an application, we reduce the power alternative for Artin groups to the power alternative for free-of-infinity Artin groups, under some conditions on their parabolic subgroups. We also introduce a uniform version of the power alternative and prove it, among other things, for a large family of two-dimensional Artin groups. As a corollary, we deduce that these Artin groups have uniform exponential growth. Finally, we prove that the power alternative is stable under taking relatively hyperbolic groups. We apply this to show that various examples, including all free-by-$\mathbb{Z}$ groups and a natural subclass of hierarchically hyperbolic groups, satisfy the uniform power alternative.

Combination theorems for Wise's power alternative

TL;DR

This work develops a unified framework of Wise's power alternative for groups acting on tree-like objects, showing WPA can be transferred from stabilisers to the whole group under a stabilisation property. It then applies this to Artin groups via visual splittings, reducing WPA to conjectural parabolic properties and proving a uniform WPA for a substantial class of (2,2)-free triangle-free Artin groups, with explicit uniform exponents and consequences such as uniform exponential growth. A parallel theory for relatively hyperbolic groups shows WPA (and uniform WPA under torsion bounds) passes from peripheral subgroups to the ambient group, with implications for acylindrical hyperbolicity and HHGs. The paper also develops a detailed set of tools (ultimate translation length, law-based variants, and mapping-tori arguments) to carry WPA through combination constructions, yielding broad new instances and several natural questions about the scope and limits of uniformity and hierarchical hyperbolicity.

Abstract

We show that Wise's power alternative is stable under certain group constructions, use this to prove the power alternative for new classes of groups, and recover known results from a unified perspective. For groups acting on trees, we introduce a dynamical condition that allows us to deduce the power alternative for the group from the power alternative for its stabilisers of points. As an application, we reduce the power alternative for Artin groups to the power alternative for free-of-infinity Artin groups, under some conditions on their parabolic subgroups. We also introduce a uniform version of the power alternative and prove it, among other things, for a large family of two-dimensional Artin groups. As a corollary, we deduce that these Artin groups have uniform exponential growth. Finally, we prove that the power alternative is stable under taking relatively hyperbolic groups. We apply this to show that various examples, including all free-by- groups and a natural subclass of hierarchically hyperbolic groups, satisfy the uniform power alternative.

Paper Structure

This paper contains 30 sections, 58 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Splittings and applications to Artin groups
  4. The uniform power alternative
  5. Relative hyperbolicity
  6. Organisation of this paper
  7. Acknowledgements
  8. Preliminaries on the Power Alternative
  9. Groups acting on trees and the power alternative
  10. Action on trees and fixed point sets
  11. Proof of Theorem \ref{['thm:wpa for action on real tree']}
  12. Examples and first application
  13. First applications
  14. Ultimate translation length
  15. Combination theorem, examples, and non-examples
  16. ...and 15 more sections

Key Result

Theorem 1

Let $T$ be a simplicial tree and let $G$ be a group acting on $T$ via graph automorphisms. Let us further assume that: Then the power alternative holds for $G$.

Figures (2)

  • Figure 1: In red and blue the fixed-point sets of $g$ and $h$, respectively. Shaded, the stable fixed-point sets.
  • Figure 2: In green, the axis of $g^n$ and its translates; in red, the axis of $h^n$ and its translates.

Theorems & Definitions (122)

  • Theorem 1: power alternative for actions on trees
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5: Theorem \ref{['thm:rel-hyp']}
  • Corollary 6: Corollary \ref{['cor:free-by-Z']}
  • Definition 2.1: power alternative
  • Lemma 2.2
  • Example 2.3
  • Lemma 2.4: stability under direct product
  • ...and 112 more