Roots of hyperelliptic involutions and braid groups modulo their center inside mapping class groups

Abstract

Let $n,k\in \mathbb{N}$ and let $S$ be the closed surface of genus $nk$. A copy of the braid group on $2k+2$ strands modulo its center is found inside $\mathrm{Mod}(S)$, provided $n\ge 3$. In particular, for $k=1$ the class of the half-twist braid inside $B_4/Z(B_4)$ is identified with a hyperelliptic involution inside $\mathrm{Mod}(S)$. As a consequence, we can show that each hyperelliptic involution inside $\mathrm{Mod}(S)$ has infinitely many square roots, and discuss their conjugacy classes. Furthermore, a copy of $\mathrm{Mod}(S_1)\cong {\mathrm{SL}}_2(\mathbb{Z})$ is found inside $\mathrm{Mod}(S_2)$. This subgroup contains the unique hyperelliptic involution on $S_2$. As a result, we can show that the latter admits infinitely many square and cubic roots, and discuss their conjugacy classes.