## Topology Seminar: Eiko Kin

- Date: 03/11/2015

*Osaka University*

University of British Columbia

Pseudo-Anosovs with small dilatations in the hyperelliptic handlebody groups and spherical Hilden groups

This is a joint work with Susumu Hirose. We consider pseudo-Anosov elements of the mapping class groups on orientable surfaces. We are interested in a numerical invariant of pseudo-Anosovs, called the dilatation. The logarithm of the dilatation of a pseudo-Anosov mapping class is called the entropy. If we fix a surface, then the set of dilatations of pseudo-Anosovs defined on the surface is closed and discrete. In particular we can talk about a minimum of any subset of dilatations defined on the surface in question.

Penner proved that the minimal entropy of pseudo-Anosovs defined on a closed surface of genus g behaves like 1/g. Later Hironaka proved that the minimal entropy of pseudo-Anosovs in the handlebody subgroup on a closed surface of genus $g$ also behaves like 1/g. We prove that the the minimal entropy of the hyperelliptic handlebody sugbroup of genus g has the same asymptotic behavior. (Our examples of pseudo-Anosovs improve the upper bound of the minimal entropy of the handlebody sugbroup given by Hironaka.) To do this, we study the spherical Hilden subgroup of the mapping class group defined on a sphere with 2n punctures, and we construct a sequence of pseudo-Anosovs with small dilatations in the spherical Hilden subgroups.

Location: ESB 4133