Classification of recurrent groups for finite-alphabet group-extended Markov systems

Let $\left(\Sigma\times G,\sigma\rtimes\Psi\right)$ be an irreducible group-extended Markov system with finite alphabet $I$, and let $\varphi:\Sigma\rightarrow\mathbb{R}$ be a Hölder-continuous potential. If $\varphi\circ\pi_{1}$ is recurrent, then $G$ is a recurrent group. Recurrent-group classification conjecture. The stated implication should hold under these hypotheses. The result is presented as following from an investigation of graph automorphisms and a theorem of Woess; its resolution is not specified in the supplied text.