Harmonic Measures on Bowditch Boundaries of Groups Hyperbolic Relative to Virtually Nilpotent Subgroups

Abstract For a group hyperbolic relative to virtually nilpotent subgroups, in a cusped graph associated to the group, we construct a reversible Markov chain whose Martin boundary is the Bowditch boundary of the group. Moreover, the harmonic measure is a conformal density corresponding to a hyperbolic Green metric and is Ahlfors regular on the Bowditch boundary. The Patterson–Sullivan density for the action on the cusped graph in this case is not in general Ahlfors-regular, but doubling, and its dimension is obtained by studying cusp excursions of geodesics. The Markov chain induces a symmetric random walk in the group, which has finite asymptotic entropy, with Martin boundary as the Bowditch boundary. Moreover, it has finite polynomial word metric-moments for suitable powers, and infinite exponential word metric-moments.