Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups

2023/06/12 by Kim, Dongryul M., Oh, Hee, Wang, Yahui · 3 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2306.06846

Abstract

Let G be a connected semisimple real algebraic group. The class of transverse subgroups of G includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup Γ, we show that the Γ-action on the Weyl chamber flow space determined by its limit set is properly discontinuous. This allows us to consider the quotient space and define Bowen-Margulis-Sullivan measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in the original spirit of Hopf-Tsuji-Sullivan. We also introduce the notion of growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint. We discuss several applications as well.

Cited by

• Classification of horospherical invariant measures in higher rank
• Kleinian viewpoints on higher rank worlds
• Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups

Related

• Patterson-Sullivan measures for transverse subgroups
• Uniqueness of conformal measures and local mixing for Anosov groups
• Ergodic dichotomy for subspace flows in higher rank
• Conformal measure rigidity and ergodicity of horospherical foliations
• Deformations of Anosov subgroups: Limit cones and Growth indicators