14 oct 2014 -- 14:00
Sala Seminari (Dip. Matematica), Pisa
Abstract.
Goldman proved that the variety Xg of conjugacy classes of representations of a surface group of genus g into PSL2R has 4g-3 connected components Xg(2-2g), ... ,Xg(2g-2) indexed by the Euler number of the representations therein. The two extremal components Xg(2-2g) and Xg(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 Xg(0) consisting of representations with Eulernumber 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.