which has become known as the *Tits-alternative* for linear groups and mapping class groups respectively.

\[tits-alt: linear\] Let $\Bbbk$ be a field, and let $\Gamma$ be a finitely generated subgroup of $GL(n,\Bbbk)$. Then, $\Gamma$ either contains a non-abelian free subgroup or is virtually solvable; moreover, these alternatives are mutually exclusive.

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

The dichotomy in the Tits alternative has been further investigated, and some refinements and variations are known for linear groups and mapping class groups; see, for example, the work of Margulis and Soĭfer on maximal subgroups of linear groups [@Margulis--Soifer:MaximalSubgrp] and the analogous result by Ivanov for mapping class groups [@Ivanov:Tits-Margulis-Soifer]. The purpose of this article is to demonstrate that the structural analogy between these groups can be reinterpreted in terms of the *girth* of finitely generated groups.

Recall that the *girth* of a graph is the length of the shortest graph cycle, if any, in the graph. In [@Schleimer:Girth], Schleimer defined the *girth* of a finitely generated group $\Gamma$ to be the 