Largest acylindrical actions of free-by-cyclic groups

TL;DR

This paper demonstrates that finitely generated free-by-cyclic groups have a largest acylindrical action on a hyperbolic space, characterizes Morse geodesics and subgroups, and computes the Morse boundary for specific cases.

Contribution

It introduces a construction of a largest acylindrical action for free-by-cyclic groups and characterizes Morse geodesics and subgroups within this framework.

Findings

Existence of a largest acylindrical action on a hyperbolic space for these groups

Characterization of Morse geodesics as those projecting to quasigeodesics in the hyperbolic space

Computation of the Morse boundary for free-by-cyclic groups with unipotent and polynomially growing monodromy

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.