groups do not satisfy any law.

It is natural to ask how the property of having infinite girth behaves under homomorphisms. The following partial answer by Akhmedov [@Akhmedov:Girth2] plays an important role in §4.

\[akhmedov criterion\] A group $\Gamma$ has infinite girth if it admits a surjection onto a non-cyclic group with infinite girth.

As noted in §\[Introduction\], we do not know if the presence of a finite-index infinite-girth subgroup of $\Gamma$ guarantees that $\Gamma$ itself has infinite girth; indeed, it is suspected that is not the case in general (see [@Akhmedov:Girth2]). This is unfortunate, because we cannot pass to a finite-index subgroup to prove that a given group has infinite girth. In our proof of the girth alternative for subgroups of mapping class groups, we go around this obstacle by applying the above proposition in a suitable manner.

Elements of Mapping Class Groups {#Elements}
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Let ${\Sigma}$ be a (not necessarily connected) compact orientable surface. A *multi-loop* on ${\Sigma}$ is a pair-wise disjoint collection of simple closed curves on ${\Sigma}$, and it is said to be *essential* if each component is not null-homotopic or peripheral.

-   An isotopy class $\calA$ of an essential (possibly empty) multi-loop on ${\Sigma}$ is 