Let $G$ be some discrete finitely generated group acting cocompactly on a leafless tree $T$. Is it true that for any natural number $n$ there is a finite graph $\Gamma$ such that: $T$ is the universal cover of $\Gamma$ and the girth of $\Gamma$ is at least $n$?



Can a tree cover a finite graph with arbitrary large girth?
Let $T$ be a tree admitting a cocompact proper action of a discrete group $G$. Then $G$ has a finite index subgroup $F$ free of finite rank. Let $(F\_n)$ be a sequence of finite index subgroups such that $\limsup F\_n=\{1\}$ (i.e. for every $g\neq 1$, eventually $g\notin F\_n$); for instance $(F\_n)$ is decreasing and $\bigcap F\_n=\{1\}$; these exist because free groups are residually finite. Then the quotients $F\_n\backslash T$ is a family of graphs covered by $T$ of girth tending to $\infty$.


On the other hand, for a leafless tree $T$ of bounded valency, to have a cocompact automorphism group is not enough: precisely a necessary and sufficient condition for $T$ to cover a finite graph is that the automorphism group of $T$ is cocompact *and* unimodular (necessity is due to Bass and Kulkarni 1990).


