Abelian subgroups of the mapping class groups for non-orientable surfaces

Abstract

Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for nonorientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free subgroup of the mapping class groups.

Injective homomorphisms of mapping class groups of non-orientable surfaces

“…Note that S(f ) = ∅ if and only if f is either pseudo-Anosov or periodic. The following was proved by Birman-Lubotzky-McCarthy [6] for orientable surfaces and by Wu [20] and Kuno [16] for non-orientable surfaces.…”

“…Theorem 2.3 (Wu [20], Kuno [16]) Let f ∈ M(N) be a non-periodic reducible mapping class. Then S(f ) = ∅, S(f ) is an adequate reduction system for f , and every adequate reduction system for f contains S(f ).…”

“…The proof for non-orientable surfaces is partially made in Wu [20] using oriented double coverings, and it is easy to get the whole result by these means. Kuno's approach [16] is different in the sense that she replaces Lemma 2.4 of [6] by another lemma of the same type.…”

“…Kuno [16] proved that the maximal rank of an abelian subgroup of M(N) is precisely rk(N). Her proof is largely inspired by Birman-Lubotzky-McCarthy [6].…”

Automorphisms of the mapping class group of a nonorientable surface

“…We define the complexity of N n g , denoted by ξ N n g , as the maximum rank of a free abelian subgroup of Mod(N n g ). By [8], for g + n > 2 we have…”