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Ergodicity of Mapping Class Group Actions on SU(2)-character varieties

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arxiv 0901.1402 v1 pith:NOWARXCL submitted 2009-01-11 math.GT

William M. Goldman , Eugene Z. Xia This is my paper · ORCID
classification math.GT
keywords groupactioncharacterclassmappingrepresentationssymplecticactions
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Let S be a compact orientable surface with genus g and n boundary components d_1,...,d_n. Let b = (b_1, ..., b_n) where b_n lies in [-2,2]. Then the mapping class group of S acts on the relative SU(2)-character variety X comprising conjugacy classes of representations f of the fundamental group F of S, where trace f(d_i) = b_i. This action preserves a symplectic structure on the open dense smooth submanifold of X. corresponding to irreducible representations. This subset has full measure and is connected. In this note we use the symplectic geometry of this space to give a new proof that this action is ergodic.

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    Potential density of integral points is established for relative SL2 and PGL2 character varieties of all smooth quasi-projective complex varieties with snc compactification.

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