Fibered faces, Penner sequences and handlebody mapping classes

Eriko Hironaka

In this paper we describe special loci on the fibered face of a 3-manifold that correspond to generalized Penner sequences of mapping classes and to handlebody mapping classes. As an application, we show that the logarithm of the minimum dilatation of handlebody mapping classes on a closed genus $g$ surface behaves asymptotically like the inverse of the genus. We also show that the minimum dilatation of mapping classes with homological dilatation equal to one shares this asymptotic behavior.