Celebratio Mathematica — Birman

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J. S. Birman, A. Lubotzky, and J. McCarthy: “Abelian and solvable subgroups of the mapping class groups,” Duke Math. J. 50 : 4 (December 1983), pp. 1107–1120. MR 726319 Zbl 0551.57004 article

Let \( M \) be an orientable, compact Riemann surface of genus \( g \) with \( b \) boundary components and \( c \) connected components. Assume each connected component of \( M \) has negative Euler characteristic. The mapping class group, \( \mathscr{M}(M) \), or \( M \) is the group of isotopy classes of orientation preserving self-homeomorphisms of \( M \), where if \( \partial M \neq \varnothing \), admissible isotopies fix each component of \( \partial M \) setwise. (Thus, in particular, the isotopy class of a Dehn twist about a curve which is parallel to a component of \( \partial M \) is considered to be trivial.) The reader is referred to [Birman 1977], [Harvey 1979], [Thurston 1988], [Fathi et al. 1975], and [Gilman 1983] for background concerning this group. The main results of this paper will be two theorems about the algebraic structure of \( \mathscr{M}(M) \):

Let \( G \) be an abelian subgroup of \( \mathscr{M}(M) \). Then \( G \) is finitely generated with torsion free rank bounded by \( 3g + b - 3c \).

Every solvable subgroup of \( \mathscr{M}(M) \) is virtually abelian. Furthermore, if \( G \) is a virtually solvable subgroup of \( \mathscr{M}(M) \), then \( G \) contains an abelian subgroup, \( A \), such that the index of \( A \) in \( G \) is bounded by \( V(M) \), where \( V(M) \) is a positive integer depending only upon \( M \).

Alexander Lubotzky	Related
John David McCarthy	Related