of ${\Sigma}_{\calC}$ and $\partial {\Sigma}_{\calC}$, and the theory is developed most naturally under the convention we adopt in this article.

Girth of Finitely Generated Groups {#Criteria}
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The *girth* of a graph is the combinatorial length of the shortest cycle in the graph if there is a nontrivial cycle in the graph, and is set to be infinity if there is no cycle in the graph. Using the girth of Cayley graphs, Schleimer introduced in [@Schleimer:Girth] the notion of the *girth* of a finitely generated group.

\[girth\] Let $\Gamma$ be a finitely generated group. Let $U(\Gamma, \calG)$ be the girth of the Cayley graph of $\Gamma$ with respect to a generating set $\calG$. The *girth* of the group $\Gamma$ is defined to be $U(\Gamma):=\sup_\calG\{U(\Gamma,\calG)\}$, where the supremum is taken over all finite generating sets $\calG$ of $\Gamma$.

Throughout the rest of the article, a group $\Gamma$ is assumed to be finitely generated unless otherwise stated.

Clearly, every finite group has finite girth and any free group has infinite girth. It is also easy to see that an abelian group has finite girth unless it is the infinite cyclic group. In this section, we discuss some criteria for the girth of 