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

TL;DR

This paper introduces the SPR property for convergence groups with GPS systems, showing that such groups admit a finite Bowen-Margulis-Sullivan measure, expanding examples beyond relatively Anosov groups.

Contribution

The paper defines the SPR property and proves that SPR groups have finite BMS measures, providing new examples in higher rank Lie groups.

Findings

SPR groups admit a finite Bowen-Margulis-Sullivan measure.

SPR property implies cocompact action on flow spaces.

Extends the class of groups with finite BMS measure beyond relatively Anosov groups.

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.