Finiteness of Bowen-Margulis-Sullivan Measure for Gromov-Patterson-Sullivan Systems

Abstract

In this paper, we develop a notion of \emph{strongly positive reccurent} (SPR) property for a convergence group with a continuous Gromov-Patterson-Sullivan (GPS) system defined by Blayac-Canary-Zhang-Zimmer. We prove that these SPR groups admits a finite Bowen-Margulis-Sullivan (BMS) measure on some associated flow spaces, which means that dynamically they admit a cocompact action on the flow spaces. This notion of SPR groups gives rise to many new examples of subgroups in higher rank Lie group that admit finite BMS measure beyond relatively Anosov groups.