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

Rou Wen

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.

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.

Paper Structure

This paper contains 18 sections, 32 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Convergence Group Action
  4. Continuous Cocycles
  5. GPS System
  6. A Flow Space
  7. Patterson--Sullivan Measure and Bowen--Margulis--Sullivan Measure
  8. Shadows
  9. Sufficient Condition for Finite BMS Measure
  10. Strongly Positive Recurrence
  11. Subset Outside of A Given Compact Set
  12. Entropy at Infinity
  13. Strongly Positively Recurrent Subgroups
  14. Examples of SPR Subgroups
  15. Schottky Products
  16. ...and 3 more sections

Key Result

Corollary 1

If there exists a compact set $K \subset \Tilde{\Omega}_{\Gamma}$ such that $\Gamma$ is positive recurrent with respect to $K$, then $m^{BMS}(\Gamma \backslash \tilde{\Omega}_{\Gamma})$ is finite. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure :

Theorems & Definitions (73)

  • Corollary : Theorem \ref{['pr implies finite']}
  • Corollary : Theorem \ref{['spr implies finite BMS']}
  • Corollary : Theorem \ref{['spr implies divergent']}
  • Definition 2.1: Convergence Group
  • Definition 2.3: Continuous Cocycles
  • Definition 2.4
  • Definition 2.6
  • Remark 2.9
  • Lemma 2.10
  • proof
  • ...and 63 more