Juan Souto

Goldman proved that the variety $X_g$ of conjugacy classes of representations of a surface group of genus $g$ into $PSL_2R$ has $4g-3$
connected components $X_g(2-2g),\dots,X_g(2g-2)$ indexed by the Eulernumber 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 ofl the other components is ergodic. I will explain why this is indeed the case the component $X_g(0)$ consisting of representations with
Eulernumber $0$ and for all $g\ge 3$. 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 $g$.