It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic manifolds with finite Bowen-Margulis measure. He uses a free product/ping-pong construction, and as such the fundamental groups of the manifolds in his examples have relatively hyperbolic fundamental groups, as of course are fundamental groups of geometrically finite hyperbolic manifolds.
My question: Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and a fundamental group which is not relatively hyperbolic?
$$\\\\\\$$ [1] Peigné, Marc, About the Poincaré exponent of a Kleinian group, Dal’Bo-Milonet, Françoise (ed.), Géométrie ergodique. Genève: L’Enseignement Mathématique (ISBN 978-2-940264-10-0/pbk). Monographie de L’Enseignement Mathématique 43, 25-59 (2013). ZBL1316.37003.
