Commensurability of geometric subgroups of mapping class groups

Abstract

Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). 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, we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.

References in corpus (1)

• Dehn twists on nonorientable surfaces

Cited by in corpus (5)

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• Crosscap slides and the level 2 mapping class group of a nonorientable surface
• On p-almost direct products and residual properties of pure braid groups of nonorientable surfaces
• On finite index subgroups of the mapping class group of a nonorientable surface
• The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$