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The boundary of hyperbolic free-by-cyclic groups

Presented by: 
Yael Algom Kfir University of Haifa
Wednesday 21st June 2017 - 10:00 to 11:00
INI Seminar Room 1
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(0) group with isolated flats and its (unique by Hruska) CAT(0)-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.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons