Quotient families of mapping classes

Eriko Hironaka

Published 2012-12-13, updated 2014-11-03Version 3

Thurston's fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperbolic 3-manifold and the monodromies of fibrations of the 3-manifold over the circle. In this paper, we generalize examples of Penner, and define quotient families of mapping classes. We show that these mapping classes correspond to open linear sections of fibered faces. The construction gives a simple way to produce families of pseudo-Anosov mapping classes with bounded normalized dilatation and computable invariants, and gives concrete examples of mapping classes associated to sequences of points tending to the interior and to the boundary of fibered faces.