# 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.