On the involution generators of the mapping class group of a punctured surface

TL;DR

This paper demonstrates that the mapping class group of a punctured surface can be generated by a small number of involutions, with specific bounds depending on the genus and number of punctures.

Contribution

It establishes new bounds on the number of involutions needed to generate the mapping class group for various punctured surfaces.

Findings

For even p ≥ 10 and g ≥ 14, three involutions suffice.

For odd p ≥ 9 and g ≥ 13, four involutions suffice.

For even p ≥ 4 and 3 ≤ g ≤ 6, four involutions suffice.

Abstract

Let Mod(Sigma_{g, p}) denote the mapping class group of a connected orientable surface of genus g with p punctures. For every even integer p \geq 10 and g \geq 14, we prove that Mod(Sigma_{g, p}) can be generated by three involutions. If the number of punctures p is odd and \geq 9, we show that Mod(Sigma_{g, p}) for g \geq 13 can be generated by four involutions. Moreover, we show that for an even integer p \geq 4 and 3 \leq g \geq 6, Mod(Sigma_{g, p}) can be generated by four involutions.