Patterson-Sullivan measures for transverse subgroups

TL;DR

This paper extends Patterson-Sullivan measure theory to transverse subgroups in higher rank Lie groups, establishing a dichotomy and analyzing subgroup critical exponents, with new results even for Anosov groups.

Contribution

It introduces Patterson-Sullivan measures for transverse subgroups, proves a Hopf-Tsuji-Sullivan type dichotomy, and derives new gap results for subgroup critical exponents.

Findings

Established Patterson-Sullivan measures for transverse groups

Proved a Hopf-Tsuji-Sullivan dichotomy for these groups

Derived new conditions for subgroup critical exponents

Abstract

We study Patterson-Sullivan measures for a class of discrete subgroups of higher rank semisimple Lie groups, called transverse groups, whose limit set is well-defined and transverse in a partial flag variety. This class of groups includes both Anosov and relatively Anosov groups, as well as all discrete subgroups of rank one Lie groups. We prove an analogue of the Hopf-Tsuji-Sullivan dichotomy and then use this dichotomy to prove a variant of Burger's Manhattan curve theorem. We also use the Patterson-Sullivan measures to obtain conditions for when a subgroup has critical exponent strictly less than the original transverse group. These gap results are new even for Anosov groups.