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The boundary of hyperbolic free-by-cyclic groups

Presented by: 
Yael Algom Kfir
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.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons