of mapping class groups $\Mod({\Sigma})$, the result was independently proved by Yamagata [@Yamagata:Girth]; see §4 for the definition of irreducible subgroups.

One difficulty in proving Theorem \[girth-alt: mcg\] is that we cannot simply pass the statement to a finite index normal subgroup; unfortunately, we don’t know that a group has infinite girth even if it has a finite index normal subgroup with infinite girth. This requires us to study the structure of $\Gamma$ carefully when ${\Sigma}$ is disconnected or when $\Gamma$ is *reducible*.

In the general context of finitely generated groups, the existence of a non-abelian free subgroup in $\Gamma$ does not imply that $\Gamma$ has infinite girth. Hence, the girth alternative is not a mere consequence of the Tits alternative. As we shall see, for subgroups of mapping class groups, the girth alternative is a slightly more intricate manifestation of underlying structural properties that are responsible for the Tits alternative. A part of the arguments in the proof of Theorem \[girth-alt: mcg\] can also be used to show the girth alternative for convergence groups (see [@Nakamura:Thesis], [@Yamagata:Girth]) and subgroups of $\Out(F_n)$ that contain an irreducible element with irreducible powers (see [@Nakamura:Thesis]).

Acknowledgement
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The results in this paper are 