contained in the Ph.D. dissertation by the author, submitted in 2008 to University of California, Davis [@Nakamura:Thesis]. The author would like to thank Dmitry Fuchs, Misha Kapovich, and his advisor Joel Hass for being the members of the dissertation committee, carefully reading the original exposition of this work, and providing insightful comments while the author was at University of California, Davis. The author would also like to thank David Futer and Igor Rivin for encouraging him to make this work publicly available in the present format.

Outline
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We present some results on the girth of finitely generated groups in §\[Criteria\]. The main tool used in the proof of Theorem \[girth-alt: mcg\] is the Infinite Girth Criterion (Proposition \[igc\]) in §\[Infinite Girth\], which reformulates and generalizes the work of Akhmedov in [@Akhmedov:Girth2]; see [@Yamagata:Girth] for a similar alternative reformulation. The essence of the proof of this criterion is a reminiscent of the classical *ping-pong argument*, which goes back to the work of Blaschke, Klein, Schottky, and Poincaré on Schottky groups in $\PSL(2,\R)$ and $\PSL(2,\C)$; see, for example, [@Klein:Ping-Pong]. In §\[Elements\] and §\[Subgroups\], we review the properties of elements and subgroups of mapping class groups from Thurston’s theory [@Thurston:Surfaces] and the 