Characterizing hierarchically hyperbolic free by cyclic groups

TL;DR

This paper provides an algebraic characterization of certain free-by-cyclic groups with coarse median properties, linking them to hierarchically hyperbolic groups and CAT(0) cube complexes.

Contribution

It introduces new algebraic conditions like 'unbranched blocks' and 'excessive linearity' to characterize hierarchical hyperbolicity in free-by-cyclic groups.

Findings

Characterization of free-by-cyclic groups with coarse median properties.

Equivalence between hierarchical hyperbolicity and being quasi-isometric to CAT(0) cube complexes.

New conditions on intersections of virtually $F_n\times \mathbb Z$ subgroups and train track representatives.

Abstract

We algebraically characterize free by cyclic groups that have coarse medians, and prove that this is equivalent to the a priori stronger properties of being colourable hierarchically hyperbolic groups and being quasi-isometric to CAT(0) cube complexes. Our algebraic characterization involves a condition on intersections between maximal virtually $F_n\times \mathbb Z$ subgroups that we call having "unbranched blocks". We also characterize hierarchical hyperbolicity of $\Gamma=F_n\rtimes_{\phi}\mathbb Z$ in terms of a property of completely split relative train track representatives of $\phi\in\mathrm{Out}(F_n)$ that we call "excessive linearity", a slight refinement of the rich linearity condition for relative train track maps introduced by Munro and Petyt.