proof of the Tits alternative [@Birman-Lubotzky-McCarthy:AbSol], [@Ivanov:Tits-Margulis-Soifer], [@McCarthy:Tits], [@Ivanov:MCGBook], and also prove a few facts that are necessary for the application of Infinite Girth Criterion in the proof of Theorem \[girth-alt: mcg\]. Finally, in §\[girth alternative\], we present the proof of Theorem \[girth-alt: mcg\].

Conventions
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For various reasons, the mapping class group $\Mod({\Sigma})$ of a connected surface ${\Sigma}$ *with boundary* is often defined in the literature to be the group of isotopy classes of $\Homeo^+({\Sigma},\partial {\Sigma})$, consisting of homeomorphisms which take each component of $\partial {\Sigma}$ to itself; that is *not* the convention we follow in this article. Our definition of $\Mod({\Sigma})$ as the group of isotopy classes of $\Homeo^+({\Sigma})$ implies that elements of $\Mod({\Sigma})$ may permute components of $\partial {\Sigma}$.

This convention coincides with the ones used in [@Birman-Lubotzky-McCarthy:AbSol], [@McCarthy:Tits], [@Ivanov:Tits-Margulis-Soifer], and [@Ivanov:MCGBook], where Thurston’s theory on mapping class groups [@Thurston:Surfaces] was utilized in the studies of the structures of subgroups. One aspect of Thurston’s theory involves a process of cutting the surface ${\Sigma}$ along an essential multi-loop which is invariant under the action of a mapping class and obtaining the induced mapping class on the resulted surface, say ${\Sigma}_{\calC}$; this new mapping class generally permutes the components 