# The boundary of hyperbolic free-by-cyclic groups

NPCW05 - Group actions and cohomology in non-positive curvature

Given an automorphism $\phi$ of the free group $F_n$ consider the HNN extension $G = F_n \rtimes_\phi \Z$. We compare two cases:
1. $\phi$ is induced by a pseudo-Anosov map on a  surface with boundary and of non-positive Euler characteristic. In this case $G$ is a CAT group with isolated flats and its (unique by Hruska) CAT -boundary is a Sierpinski Carpet (Ruane).
2. $\phi$ is atoroidal and fully irreducible. Then by a theorem of Brinkmann $G$ is hyperbolic. If $\phi$ is irreducible then Its boundary is homeomorphic to the Menger curve (M. Kapovich and Kleiner).
We prove that if $\phi$ is atoroidal then its boundary contains a non-planar set. Our proof highlights the differences between the two cases above.
This is joint work with A. Hilion and E. Stark.

This talk is part of the Isaac Newton Institute Seminar Series series.