Generating the Mapping Class Group of a Punctured Surface by Involutions

Abstract

Let Σ g,b denote a closed orientable surface of genus g with b punctures and let Mod(Σ g,b ) denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, Mod(Σ g,b ) is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate Mod(Σ g,b ). Brendle and Farb [BF] gave an answer in the case of g ≥ 3, b = 0 and g ≥ 4, b = 1, by describing a generating set consisting of 6 involutions. Kassabov showed that for every b Mod(Σ g,b ) can be generated by 4 involutions if g ≥ 8, 5 involutions if g ≥ 6 and 6 involutions if g ≥ 4. We proved that for every b Mod(Σ g,b ) can be generated by 4 involutions if g ≥ 7 and 5 involutions if g ≥ 5.

On groups generated by involutions of a semigroup

“…Involutions are ubiquitous in many branches of mathematics, and have played a particularly significant role in algebra. There are algebras that have (external) involution operators defined on them [3,8,17,43,47,50,60,67,79,82,86,90,97], as well as algebras generated by (internal) involutions such as the well-known Coxeter Groups, mapping class groups, special linear groups, and non-abelian finite simple groups [15,20,44,52,63,64,78,100]. An (internal) involution in a group is an element of order 2 (i.e., a non-identity element a that satisfies a 2 = 1).…”