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Geometry and Dynamics of Transverse Groups

Richard Canary, Tengren Zhang, Andrew Zimmer

Abstract

We survey recent work on the geometry and dynamics of transverse subgroups of semi-simple Lie groups.

Geometry and Dynamics of Transverse Groups

Abstract

We survey recent work on the geometry and dynamics of transverse subgroups of semi-simple Lie groups.

Paper Structure

This paper contains 11 sections, 30 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Fuchsian groups
  3. Divergent and transverse groups
  4. Patterson--Sullivan measures for divergent groups
  5. The Shadow Lemma for transverse groups
  6. Flow spaces for transverse groups
  7. Recent work
  8. Kim, Oh and Wang's approach to dynamics of transverse group
  9. A general theory of Patterson-Sullivan measures for convergence group actions
  10. Applications to transverse and relatively Anosov groups
  11. Other work

Key Result

Theorem 1.1

If $X$ is a closed hyperbolic surface, then the geodesic flow on $T^1X$ is ergodic with respect to the Liouville measure, i.e. if $A\subset T^1X$ is a measurable set which is invariant under the flow, then $A$ either has zero or full Liouville measure.

Theorems & Definitions (36)

  • Theorem 1.1: Hedlund hedlund
  • Theorem 1.2: Huber huber
  • Theorem 2.1: Patterson patterson
  • proof : Sketch of proof:
  • Lemma 2.2: Patterson
  • Theorem 2.3: Sullivan's Shadow Lemma sullivan-density
  • proof : Sketch of proof
  • Corollary 2.4
  • proof : Sketch of proof
  • Theorem 2.5: Brooks brooks
  • ...and 26 more