Georgia Institute of Technology College of Sciences

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 openquestion; one particularly interesting case is SL(n,Z). Thurston conjectured that SL(n,Z) has a quadratic Dehn function when n>=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>=5, and in this talk, I will give an introduction to the Dehn function, discuss some of the background of the problem and, time permitting, give a sketch of the proof.