  We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of
a surface of genus $g$ with $n$ punctures, by the subgroup
$\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our
first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$
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 $\operatorname{Mod}_g^0/
\operatorname{Mod}_g^0[p]$ contains a K\"ahler subgroup of finite index, for
every $p\ge 2$ coprime with six. Finally, we observe that the existence of
finite-index subgroups of $\operatorname{Mod}_g^0$ with infinite abelianization
is equivalent to the analogous problem for $\operatorname{Mod}_g^0/
\operatorname{Mod}_g^0[p]$.
