Geometry and Topology Seminar

Fibered faces of hyperbolic 3-manifolds partition the set of
pseudo-Anosov mapping classes on surfaces into families with similar
dynamical properties. As a consequence, dilatations normalized by Euler
characteristic of the underlying surface vary continuously as you move
along on a fibered face, and the dilatations themselves are the zeros of
specializations of a single multi-variable polynomial, called the
Teichmueller polynomial.

Dowdall-Kapovich-Leininger developed a version of fibered face theory for
free-by-cyclic groups defined by free group outer automorphisms that admit
expanding train track maps. They define convex open cones in the BNS
invariant for the free-by-cyclic group whose primitive integral points
correspond to deformations of the original automorphism. They also show
that normalized dilatations vary continuously.

Independently Dowdall-Kapovich-Leininger and AlgomKfir-Hironaka-Rafi
defined an analog of the Teichmueller polynomial associated to the DKL
cone that computes dilatations. The Teichmueller polynomial in turn
defines a typically larger convex polygonal cone, which
Dowdall-Kapvoich-Leininger show is a connected component of the BNS
invariant of the associated free-by-cyclic group defined by the original
automorphism.

In the first half of this talk, we review Thurston's fibered face theory,
and BNS theory in the context of free-by-cycilc groups. In the second, we
give the AlgomKfir-Hironaka-Rafi definition of the Teichmueller polynomial
as an invariant of a 2-complex with semiflow, using data encoded by a
digraph and cycle complex. If time allows, we will relate these results to
McMullen's recent work on the size of Perron numbers.