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The visual boundary of hyperbolic free-by-cyclic groups

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Emily Stark (Technion)

Abstract: Given an automorphism of the free group, we consider the mapping
torus defined with respect to the automorphism. If the automorphism is
atoroidal, then the resulting free-by-cyclic group is hyperbolic by work
of Brinkmann. In addition, if the automorphism is fully irreducible, then
work of Kapovich--Kleiner proves the boundary of the group is homeomorphic
to the Menger curve. However, their proof is very general and gives no
tools to further study the boundary and large-scale geometry of these
groups. In this talk, I will explain how to construct explicit embeddings
of non-planar graphs into the boundary of these groups whenever the group
is hyperbolic. Along the way, I will illustrate how our methods
distinguish free-by-cyclic groups which are the fundamental group of a
3-manifold. This is joint work with Yael Algom-Kfir and Arnaud Hilion.