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

TL;DR

This work investigates hyperelliptic involutions in mapping class groups by constructing a Dehn-twist–based framework that yields infinite square roots (and sometimes cubic roots) of these involutions. It identifies explicit embeddings of $\mathrm{SL}_2(\mathbb{Z})$ inside $\mathrm{Mod}(S_2)$ and, for genus $g\ge3$, embeds braid groups modulo their centers $B_{2k+2}/Z(B_{2k+2})$ inside $\mathrm{Mod}(S)$ via the groups $\Theta_n^k$. A central contribution is proving $\Theta_n^k \cong B_{2k+2}/Z(B_{2k+2})$ for $n\ge3$, and expressing hyperelliptic involutions as products like $(abc)^2$, linking to half-twists in braids; the paper also analyzes roots and their conjugacy in low genera and provides a lattice-based classification framework for all $\Theta_n^k$. These results reveal deep connections between hyperelliptic involutions, braid groups, and linear groups within mapping class groups, with explicit constructions enabling further study of roots and conjugacy classes.

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 $n\geq 3$. 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.

Figures (10)

• Figure 1:
• Figure 2: The classification of groups of type $\Theta_n^k$.
• Figure 3:
• Figure 4:
• Figure 5: Each half-twist $H_{\gamma_i}$ lifts to a Dehn twist $T_{\overline{\gamma}_i}.$

Theorems & Definitions (18)

• Theorem 2.1
• proof
• Theorem 3.1: the Birman-Hilden theorem for $S_k^2$
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• proof