Word measures on unitary groups

Monday 13th November, 2017 15:00-16:00 Maths 311B

This is joint work with Doron Puder (Tel Aviv University).

For a positive integer r, fix a word w in the free

group on r generators. Let G be any group. The word

w gives a `word map' from G^r to G: we simply replace the

generators in w by the corresponding elements of G. We

again call this map w. The push forward of Haar measure under

w is called the w-measure on G. We are interested in

the case G = U(n), the compact Lie group of n-dimensional

unitary matrices. A motivating question is: to what extent do the

w-measures on U(n) determine algebraic properties of the

word w?

We proved in our first paper that one can detect the

'stable commutator length' of w from the w-measures on

U(n). Our main tool was a formula for the Fourier

coefficients of w-measures; the coefficients are rational

functions of the dimension n, for reasons coming from

representation theory.

We can now explain all the Laurent coefficients of these

rational functions in purely topological terms. I'll explain all

this in my talk, which should be broadly accessible and of general

interest. I'll also invite the audience to consider some

remaining open questions.

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