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

Abstract

One of the basic and important problems to study algebraic structures of the mapping class groups is finding abelian subgroups included in the mapping class groups. Birman-Lubotzky-McCarthy gave the answer of this question for the orientable surfaces, namely, they proved that any abelian subgroup of the mapping class groups for orientable surfaces of genus $g$ with $b$ boundary components and $c$ connected components is finitely generated and the maximal torsion-free rank of it is $3g+b-3c$. In the present paper, we prove that any abelian subgroup of the mapping class group of a compact connected non-orientable surface $N$ of genus $g\geq 1$ with $n\geq 0$ boundary components whose Euler characteristic is negative is finitely generated and the maximal torsion-free rank of it is $\frac{3}{2}(g-1)+n-2$ if $g$ is odd and $\frac{3}{2}g+n-3$ if $g$ is even.