## Topology and Geometry Seminar

Title: Pseudo-Anosov mapping classes with small dilatation from graphs

Speaker: Eriko Hironaka

Abstract:
Heuristically there are reasons to think that mapping classes with
small complexity should arise from simple graphs. We describe two
ways to construct pseudo-Anosov maps from graphs, and relate their
dilatations to spectral radius of the adjacency matrix.
The first construction is due to Thurston, and is a way to
construct orientable pseudo-Anosov maps from bipartite graphs.
The second construction uses labeled graphs. A particular case
was used in work with E. Kin to give an explicit of a family
of mapping classes with dilatations giving the best known
asymptotics as a function of genus.
Together these two families give the smallest known dilatations
for mapping classes of low genus.