## Osaka Journal of Mathematics

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

Erika Kuno

#### 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.

#### Article information

Source
Osaka J. Math., Volume 56, Number 1 (2019), 91-100.

Dates
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.ojm/1547607628

Mathematical Reviews number (MathSciNet)
MR3908779

Zentralblatt MATH identifier
07055401

Subjects
Primary: 20F38: Other groups related to topology or analysis
Secondary: 20K27: Subgroups

#### Citation

Kuno, Erika. Abelian subgroups of the mapping class groups for non-orientable surfaces. Osaka J. Math. 56 (2019), no. 1, 91--100. https://projecteuclid.org/euclid.ojm/1547607628

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