supremum of the girth of Cayley graphs of $\Gamma$ over all finite generating sets. By the observations of Schleimer in [@Schleimer:Girth] and the subsequent work of Akhmedov in [@Akhmedov:Girth1] and [@Akhmedov:Girth2], there appears to be a significant qualitative difference between groups with finite girth and groups with infinite girth. In particular, Akhmedov gave the following “*girth alternative”* for linear groups, which shows that the division essentially coincides with the dichotomy in the Tits alternative for linear groups.

\[[@Akhmedov:Girth2]\] \[girth-alt: linear\] Let $\Bbbk$ be a field, and let $\Gamma$ be a finitely generated subgroup of $GL(n,\Bbbk)$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually solvable group; moreover, these alternatives are mutually exclusive.

Our main result is the following analogous girth alternative for mapping class groups $\Mod({\Sigma})$, showing that the division of subgroups into the ones with finite girth and the ones with infinite girth again coincides with the dichotomy in the Tits alternative.

\[girth-alt: mcg\] Let ${\Sigma}$ be a compact orientable surface, and let $\Gamma$ be a finitely generated subgroup of $\Mod({\Sigma})$. Then, $\Gamma$ is either a non-cyclic group with infinite girth or a virtually abelian group; moreover, these alternatives are mutually exclusive.

For *irreducible* subgroups 