skip to main content

Geometry and Topology Seminar

Friday, February 20, 2015
3:00pm to 5:00pm
Add to Cal
Fibered face theory and entropy for free-by-cyclic groups
Eriko Hironaka, Professor, Mathematics, Florida State University,

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

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.

For more information, please contact Maria Trnkiova by email at [email protected] or visit