Quotients of the mapping class group by power subgroups

Abstract

We study the quotient of the mapping class group ${\mathrm{Mod}}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup ${\mathrm{Mod}}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that ${\mathrm{Mod}}_g^1/{\mathrm{Mod}}_g^1[p]$ contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher-rank lattice, for all but finitely many explicit values of $p$. Next, we prove that ${\mathrm{Mod}}_g^0/{\mathrm{Mod}}_g^0[p]$ contains a Kähler subgroup of finite index, for every $p\ge 2$ coprime with six. Finally, we observe that the existence of finite-index subgroups of ${\mathrm{Mod}}_g^0$ with infinite abelianization is equivalent to the analogous problem for ${\mathrm{Mod}}_g^0/{\mathrm{Mod}}_g^0[p]$.

corrected version

References in corpus (3)

• Quotients of surface groups and homology of finite covers via quantum representations
• On finiteness properties of the Johnson filtrations
• Profinite completions of Burnside-type quotients of surface groups

Cited by in corpus (1)

• A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups