The torsion generating set of the mapping class groups and the Dehn twist subgroups of non-orientable surfaces of odd genus (Q2203646)

The basic problem considered in the present paper is to find a simple generating set for the mapping class group of a closed non-orientable surface \(N_g\) of genus \(g\), reducing both the number and the orders of the generators. In a previous paper [J. Knot Theory Ramifications 26, No. 11, Article ID 1750061, 9 p. (2017; Zbl 1386.57026)], the author showed that the extended mapping class group of an orientable surface of genus \( g \ge 5\) can be generated by two elements of finite orders 2 and \(4g+2\) (this remains true also for genus 3 and 4). In the present paper, the author applies his methods to the mapping class groups of non-orientable surfaces of odd genus, and also to their subgroups of index 2 generated by all Dehn twists (the ``twist subgroups''). The main result is the following. If \(g = 4k+3\) and \(k\ge 1\) then the mapping class group of a non-orientable surface \(N_g\) of genus \(g\) can be generated by three elements of finite orders 2, 2 and \(2g\); if \(g = 4k+1\) and \(k \ge 2\) then the twist subgroup of index 2 can again be generated by three elements of finite orders 2, 2 and \(2g\). \textit{B. Szepietowski} proved [Geom. Dedicata 117, 1--9 (2006; Zbl 1091.57014)] that the mapping class group of \(N_g\), for \(g \ge 4\), can be generated by three elements or by four involutions.