Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Oh, Hee

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 relatively Anosov subgroups. Given a transverse subgroup Gamma, we show that the Gamma-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 study the ergodicity of a one-parameter diagonalizable subgroup of G acting on the aforementioned quotient space, equipped with a Bowen-Margulis-Sullivan measure. We obtain an ergodicity criterion similar to the Hopf-Tsuji-Sullivan dichotomy for the ergodicity of the geodesic flow on hyperbolic manifolds. In addition, we extend this criterion to the action of any connected diagonal subgroup of arbitrary dimension. This provides a codimension dichotomy on the ergodicity of a connected diagonalizable subgroup for general Anosov subgroups, generalizing an earlier work by Burger-Landesberg-Lee-Oh for Borel Anosov subgroups. 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. Moreover, we explore its applications to general Anosov subgroups and obtain some correspondence on conformal measures similar to an earlier work by Lee-Oh for Borel Anosov subgroups.

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