We give a sufficient condition for the girth of finitely generated groups to be infinite by using a technique to prove a so-called ping-pong lemma or table tennis lemma. We show that some convergence groups and subgroups of mapping class groups satisfy the condition. Therefore the girth of each of them is infinite.
"The girth of convergence groups and mapping class groups." Osaka J. Math. 48 (1) 233 - 249, March 2011.