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PATCH Day at Temple

September 21, 2012

Abstracts

Thomas Koberda, The complex of curves for a right-angled Artin group

ABSTRACT: I will discuss an analogue of the curve complex for right-angled Artin groups and describe some of its properties. I will then show how it guides parallel results between the theory of mapping class groups and the theory of right-angled Artin groups. Joint with Sang-hyun Kim.

Eriko Hironaka, Small dilatation pseudo-Anosovs

ABSTRACT: A pseudo-Anosov mapping classes on a compact finite-type oriented surface S has the property that the growth rate of lengths of an essential simple closed curve under iterations of the mapping class is exponential, and the growth rate is independent of the choice of curve and the of the choice of metric. This growth rate is called that dilatation of the mapping class. In this talk, we discuss the problem of describing small dilatation pseudo-Anosov mapping classes, i.e., those such that the dilatation raised to the topological Euler characteristic of the surface is bounded. We describe small dilatation mapping classes in terms of deformations within fibered faces, and give some explicit examples. We finish the talk with a conjecture concerning the "shape" of small dilatation mapping classes.

Andrew Putman, Stability in the homology of congruence subgroups

ABSTRACT: I'll discuss some recent results which uncover new patterns in the homology groups of congruence subgroups of $SL_n(\mathbb{Z})$ and related groups.