Word measures on unitary groups

Michael Magee (Durham)

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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