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The University of Southampton
Mathematical Sciences

Research project: Finite subgroups of mapping class groups of closed surfaces

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The mapping class group MCG(S) of a closed orientable surface S of genus g (at least two) is the group of (orientation-preserving) homeomorphisms of S, modulo homotopy. Mapping class groups arise in many contexts, and provide a core set of examples in geometric group theory. Mapping class groups are well studied, and for example it is well known that MCG(S) is finitely generated, that a finite subgroup of MCG(S) has order at most the Hurwitz bound of 84(g-1), and that the number of such finite subgroups (up to conjugacy) is finite.

The goal of this project, which is joint work with Dr Aaron Wootton (University of Portland, Portland OR, USA) is to bound the number of such finite subgroups in terms of g. We have a quadratic lower bound, obtained using a counting argument on the skeletal signature space.

Papers

J W Anderson and A Wootton, A Lower Bound for the Number of Group Actions on a Compact Riemann Surface, Algebraic and Geometric Topology 12 (2012), 19--35, arXiv:1107.3433

Related research groups

Pure Mathematics
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