Ergodicty of the mapping class group action on a component of the character variety

Juan Souto (Université de Rennes I)

Monday 29th September, 2014 16:00-17:00 Maths 326

Abstract

Goldman proved that the variety X_g of conjugacy classes
of representations of a surface group of genus g into PSL₂R has
4g-3 connected components X_g(2-2g), ... ,X_g(2g-2) indexed by the
Euler number of the representations therein. The two extremal
components X_g(2-2g) and X_g(2g-2) correspond to Teichmueller
spaces on which the mapping class group acts discretely. On the other
hand Goldman conjectured that the action on each one of the other
components is ergodic. I will explain why this is indeed the case the
component X_g(0) consisting of representations with Euler number 0
and for all g ≥ 3. The basic technical result is a formula relating
the euler number of a representation and the infimum of the energies
of equivariant harmonic maps where the infimum is taken over all maps
and all conformal structures on the surface of genus g.

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