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Largest acylindrical actions of free-by-cyclic groups

Monika Kudlinska, Harry Petyt

TL;DR

The paper establishes a canonical largest acylindrical action for finitely generated free-by-cyclic groups by coning off maximal product subgroups, yielding a hyperbolic model $X$ on which $G$ acts acylindrically and non-elementarily. It proves that Morse geodesics in $G$ are precisely the projections of quasigeodesics in $X$, giving Morse local-to-global for these groups and enabling a universal recognition framework for stable subgroups. For UPG monodromy and its polynomial growth, the authors construct a quasitree model and derive acylindrical actions, leading to a detailed analysis of stability, strong quasiconvexity, and the Morse boundary. The results unify several aspects of non-positive curvature obstructions in free-by-cyclic groups and provide a robust toolkit for understanding their quasiconvex subgroups and Morse boundaries, with implications for hierarchical structures and relative hyperbolicity.

Abstract

We show that every finitely generated free-by-cyclic group $G$ admits a largest acylindrical action on a hyperbolic space $X$ obtained by coning off maximal product subgroups of $G$. We characterise Morse geodesics of $G$ as those that project to quasigeodesics in $X$, thus showing that all finitely generated free-by-cyclic groups are Morse local-to-global. We also characterise the stable and strongly quasiconvex subgroups of $G$. Finally, we compute the Morse boundary for \{finitely generated free\}-by-cyclic groups with unipotent and polynomially growing monodromy.

Largest acylindrical actions of free-by-cyclic groups

TL;DR

The paper establishes a canonical largest acylindrical action for finitely generated free-by-cyclic groups by coning off maximal product subgroups, yielding a hyperbolic model on which acts acylindrically and non-elementarily. It proves that Morse geodesics in are precisely the projections of quasigeodesics in , giving Morse local-to-global for these groups and enabling a universal recognition framework for stable subgroups. For UPG monodromy and its polynomial growth, the authors construct a quasitree model and derive acylindrical actions, leading to a detailed analysis of stability, strong quasiconvexity, and the Morse boundary. The results unify several aspects of non-positive curvature obstructions in free-by-cyclic groups and provide a robust toolkit for understanding their quasiconvex subgroups and Morse boundaries, with implications for hierarchical structures and relative hyperbolicity.

Abstract

We show that every finitely generated free-by-cyclic group admits a largest acylindrical action on a hyperbolic space obtained by coning off maximal product subgroups of . We characterise Morse geodesics of as those that project to quasigeodesics in , thus showing that all finitely generated free-by-cyclic groups are Morse local-to-global. We also characterise the stable and strongly quasiconvex subgroups of . Finally, we compute the Morse boundary for \{finitely generated free\}-by-cyclic groups with unipotent and polynomially growing monodromy.

Paper Structure

This paper contains 15 sections, 29 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Further questions
  3. Preliminaries
  4. Bass--Serre theory
  5. Dynamics of outer automorphisms
  6. Free-by-cyclic groups
  7. Splittings of UPG free-by-cyclic groups
  8. Acylindrical actions on trees
  9. The UPG case
  10. The construction
  11. Quasitree and acylindricity
  12. The general case
  13. Morseness and stability
  14. Stable and strongly quasiconvex subgroups
  15. Morse boundary

Key Result

Theorem 1.1

Let $G$ be a free-by-cyclic group with a finite generating set $S$, and let $\mathcal{P}$ be the collection of maximal product subgroups. The Cayley graph $X = \mathrm{Cay}(G; S \cup \mathcal{P})$ is hyperbolic, and it is unbounded if $G$ is not virtually a product. The regular action of $G$ on $X$

Figures (1)

  • Figure 1: The construction of the path $p_{xy}$ in the proof of Proposition \ref{['prop:quasitree']}. The sequence of $X_{v_{i_j}}$ is in bold. Starting with $i_0=0$, we have $i_0^+=2$ because $v_2\in\bar{N}(v_0)$, and then $i_1=3$. Next $i_1^+=4$, and $i_2=6$ because $v_4\in\bar{N}(v_6)$. The subspace $\bar{X}_{v_{i_1}}$ is highlighted in purple. The points $x_1$ and $y_1$ are produced by projecting to it.

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1: bestvinafeighnhandel:tits:1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • ...and 40 more