inductively that any nontrivial reduced word in $\sigma^{\pm 1}$ and $\tau^{\pm 1}$ takes $x \in X-(U_\sigma \cup U_\tau)$ into $U_\sigma \cup U_\tau$ via the action of $\langle \sigma, \tau \rangle$, showing that the word cannot represent the identity element of $\Gamma$. This is the classical *ping-pong argument*.

In the study of the girth of groups, a criterion for a group to have infinite girth is instrumental. Generalizing and reformulating the work of Akhmedov [@Akhmedov:Girth2], we give such a criterion in comparable generality as Proposition \[fsc\]; see [@Yamagata:Girth] for a similar alternative reformulation.

\[igc\] Let $\Gamma$ be a group acting on a set $X$, with a finite generating set $\calG:=\{\gamma_1, \cdots , \gamma_n\}$. Suppose there exist elements $\sigma, \tau \in \Gamma$, subsets $U_\sigma, U_\tau \subset X$, and a point $x \in X$, such that

1.  $\displaystyle{x \not \in (U_\sigma \cup U_\tau) \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma_i^\varepsilon(U_\sigma \cup U_\tau)}$,

2.  $\displaystyle{\sigma^k \bigg( \{x\} \cup U_\tau \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma_i^\varepsilon(U_\tau) \bigg) \subset U_\sigma }$ for all $k \in \Z-\{0\}$, and

3.  $\displaystyle{\tau^k \bigg( \{x\} \cup U_\sigma \cup \bigcup_{\varepsilon=\pm1} \bigcup_{i=1}^n \gamma_i^\varepsilon(U_\sigma) \bigg) \subset U_\tau }$ for all $k \in \Z-\{0\}$.

Then, $\Gamma$ is a non-cyclic group with infinite girth.

Clearly, $\sigma$, $\tau$, $U_\sigma$, $U_\tau$, and 