Abstract
Let M be an orientable surface with punctures and/or boundary components. Paris and Rolfsen (J Reine Angew Math 521:47–83, 2000) studied “geometric subgroups” of the mapping class group of M, that is subgroups corresponding to inclusions of connected subsurfaces. In the present paper we extend this analysis to disconnected subsurfaces and to the nonorientable case. We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, following the work of Paris (Math Ann 322:301–315, 2002), we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.
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Stukow, M. Commensurability of geometric subgroups of mapping class groups. Geom Dedicata 143, 117–142 (2009). https://doi.org/10.1007/s10711-009-9377-y
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DOI: https://doi.org/10.1007/s10711-009-9377-y