Monika Kudlinska

Hyperbolicity in free-by-cyclic groups revisited

A group G is free-by-cyclic if it admits a map onto ℤ with free kernel. By combining the work of Brinkmann with recent results of Linton, it follows that a finitely generated free-by-cyclic group is Gromov hyperbolic exactly when it does not contain non-trivial product subgroups. I will discuss recent joint work with Harry Petyt where we prove a geometric version of the Brinkmann–Linton result. More precisely, we show that for any f.g. free-by-cyclic group G, the relative Cayley graph of G with respect to all product subgroups is hyperbolic and the induced G-action is acylindrical. We also show that the relative Cayley graph detects Morse quasigeodesics of G and thus stable and strongly quasiconvex subgroups. As a corollary, we obtain that all f.g. free-by-cyclic groups are Morse local-to-global and admit universal recognising spaces (in the sense of Balasubramanya–Chesser–Kerr–Mangahas–Trin) with largest acylindrical actions. Such behaviour is analogous to the setting of right-angled Artin groups, mapping class groups, and more generally, hierarchically hyperbolic groups, as well as the outer automorphism groups of free groups.