Characterizing hierarchically hyperbolic free by cyclic groups

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 $\hat{I}“=F_n{⋊}_{\ddot{I}}•\mathbb{Z}$ in terms of a property of completely split relative train track representatives of $\ddot{I}•\in \mathrm{Out}(F_n)$ that we call "excessive linearity", a slight refinement of the rich linearity condition for relative train track maps introduced by Munro and Petyt.

Introduction and some sections were expanded and restructured. Comments are welcome!