Ergodic theory on moduli spaces. (English) Zbl 0907.57009

The paper is concerned with closed orientable surfaces \(M\) of genus \(> 1\). The mapping class group of \(M\) is the \(\pi_0\) of the groups of diffeomorphisms of \(M\). A result by J. Nielsen states that the mapping class group is isomorphic to \(Out(\pi)\). For a Lie-group \(G\) let \(\operatorname{Hom}(\pi,G)/G\) be the space of \(G\) orbits on the set of homomorphisms from \(\pi\) to \(G\). Geometrically this space can be seen as the moduli space of flat principal \(G\)-bundles on \(M\). By the isomorphism of Nielsen mentioned above the mapping class group acts on \(\operatorname{Hom}(\pi,G)/G\) and on the spaces \(\operatorname{Hom}_\xi(\pi,G)/G\) twisted by an element \(\xi\) from the center of \(G\). If \(G\) is semistable the action preserves the symplectic structure and therefore the volume form [the author, J. Reine Angew. Math. 407, 126-159 (1990; Zbl 0692.53015)]. The symplectic structure on the compact, singular space \(\operatorname{Hom}(\pi,G)/G\) is singular along the singular stratum. By a result of J. Huebschmann [Duke Math. J. 80, No. 3, 737-756 (1995; Zbl 0852.58037)] the measure of the space is nonetheless finite. The first main result of the paper states that if \(G\) is a compact Lie-group that is locally isomorphic to \(SU(2)\) or \(U(1)\) then for every \(\xi\) in the center of \(G\) the action of the mapping class group on \(\operatorname{Hom}_\xi(\pi,G)/G\) is ergodic. Moreover, it is conjectured that the conclusion of the theorem is true for all compact Lie-groups. The first main result is included in the more general result that under the same hypothesis the mapping class group acts ergodically on the relative character variety of the surface. Several other more technical results centering around the same topic are provided.
Reviewer: V.Welker (Essen)


57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
58D29 Moduli problems for topological structures
57M05 Fundamental group, presentations, free differential calculus
22D40 Ergodic theory on groups
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