Geometry Seminar: Characterizing hierarchically hyperbolic free by cyclic groups

Hierarchical hyperbolicity is a generalization of hyperbolicity that aims to capture key features of not-quite-hyperbolic spaces and groups, perhaps most notably the mapping class group of a surface. This talk concerns free-by-cyclic groups, i.e., mapping tori of outer automorphisms of finite-rank free groups. The principal result provides the first complete algebraic characterization of when a free-by-cyclic group admits a hierarchically hyperbolic structure by using a transparent condition on intersections between particular maximal subgroups. We describe this as having "unbranched blocks" and show that it is equivalent to a host of a priori stronger geometric properties, such as being quasi-isometric to a finite-dimensional CAT(0) cube complex.

This represents joint work with Mark Hagen, Funda Gültepe, and Pritam Ghosh.

The talk is intended to be self-contained and accessible to a general topologically- or geometrically-inclined audience. Knowledge of hyperbolic geometry will not be assumed, although foundational facts will be presented without proof.