Recurrence and pressure for group extensions

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterises recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincar\'e series of Kleinian groups and the cogrowth of group presentations.