It is well-known that if $\widetilde M\to M$ is a Galois cover of a compact Riemannian manifold $M$ with deck-transformation group $G$, then the growth of $G$ equals the volume growth of $\widetilde M$ (in the pullback metric).
Question. Is the same true when $M$ is a finite volume complete Riemannian manifold and $G$ is finitely generated?
The usual proof (a la Svarc-Milnor) argues that $G$ and $\widetilde M$ are quasiisometric (since $M$ is compact), and then uses that growth is a quasiisometry invariant. One could hope that the reasoning extends to the finite volume case when quasi-isometry is replaced by measure equivalence (ME), but growth type is not an invariant of ME. On the other hand, I do not have counterexamples for the above question.