# Geometric Analysis and Topology Seminar

#### The Dehn Function of SL(n;Z)

Speaker: Robert Young, NYU and Toronto

Location: Warren Weaver Hall 202

Date: Friday, October 29, 2010, 11 a.m.

Synopsis:

The Dehn function is a group invariant which connects geometric and combinatorial group theory; it measures both the difficulty of the word problem and the area necessary to fill a closed curve in an associated space with a disc. The behavior of the Dehn function for high-rank lattices in high-rank symmetric spaces has long been an open question; one particularly interesting case is $$\mathrm{SL}(n;\mathbb{Z})$$. Thurston conjectured that $$\mathrm{SL}(n;\mathbb{Z})$$ has a quadratic Dehn function when $$n \geq 4$$. This differs from the behavior for $$n = 2$$ (when the Dehn function is linear) and for $$n = 3$$ (when it is exponential). I have proved Thurston's conjecture when $$n \geq 5$$, and in this talk, I will give an introduction to the Dehn function, discuss some of the background of the problem and give a sketch of the proof.