a group to be finite or infinite in a general setting. The Infinite Girth Criterion (Proposition \[igc\]) in §\[Infinite Girth\] is employed as the main tool in §\[girth alternative\].

Criteria for Finite Girth {#Criteria: Finite Girth}
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Recall that a group $\Gamma$ is said to satisfy a law if there is a word $w(x_1, \cdots , x_n)$ on $n$ letters such that $w(\gamma_1, \cdots , \gamma_n)=1$ in $\Gamma$ for any $\gamma_1, \cdots , \gamma_n \in \Gamma$. Schleimer obtained an important criterion for a group to have finite girth in [@Schleimer:Girth].

\[law criteria\] A group $\Gamma$ has finite girth if it satisfies a law and is not infinite cyclic.

Together with the proposition below, Theorem \[law criteria\] provides a significant portion of the class of groups that are known to have finite girth.

\[schleimer criteria\] A group $\Gamma$ has finite girth if it satisfies one of the following conditions:

-   $\Gamma$ contains a finite-index subgroup with finite girth.

-   $\Gamma$ admits a finite-kernel surjection onto a group with finite girth.

\[virtually solvable\] A virtually solvable group $\Gamma$ has finite girth, unless it is infinite cyclic.

One may ask if amenable groups, other than the infinite cyclic one, always have finite girth. 