\titleofpresentation{Commutator maps, measure preservation and character 
methods}

\presenter{Aner Shalev}

Given a group $G$ we study the commutator map $f\colon G\times G\to G$
where $f(x,y)=x^{-1}y^{-1}xy$. When $G$ is finite and has very
few representations in some sense, we show that the map $f$
is almost measure preserving. In particular almost all
elements of such groups are commutators.

The main applications of this technique are for finite
simple groups. We prove results towards a longstanding conjecture
of Ore (which is still open), and confirm a conjecture of Guralnick
and Pak on the Product Replacement Algorithm. This is joint work
with Shelly Garion.

I will also mention recent results with Liebeck and O'Brien on
Ore's conjecture, and with Larsen on more general word maps
and Waring type problems.

Our methods combine representation theory with probabilistic
arguments. In particular the so called Witten zeta function
encoding character degrees plays a key role in the proofs.

\presenteraddress{Hebrew University}

\email{shalev@math.huji.ac.il}
