Invariant Radon measures for Unipotent flows and products of Kleinian groups

TL;DR

This paper classifies all ergodic, conservative, invariant Radon measures for the diagonal horospherical subgroup action on a space formed by products of Kleinian groups, extending Benoist-Quint's work to new settings.

Contribution

It provides a classification of invariant Radon measures for horospherical actions on product spaces involving Kleinian groups, under geometric finiteness assumptions.

Findings

Classified ergodic, conservative, invariant Radon measures for N-action.

Extended Benoist-Quint classification to new group actions.

Applied to spaces with geometrically finite Kleinian groups.

Abstract

Let G=PSL(2, F) where F= R or C, and consider the space Z=(\Gamma_1 x \Gamma_2)\ (G x G) where \Gamma_1<G is a co-compact lattice and \Gamma_2<G is a finitely generated discrete Zariski dense subgroup. The work of Benoist-Quint gives a classification of all ergodic invariant Radon measures on Z for the diagonal G-action. In this paper, for a horospherical subgroup N of G, we classify all ergodic, conservative, invariant Radon measures on Z for the diagonal N-action, under the additional assumption that \Gamma_2 is geometrically finite.