Patterson-Sullivan measures of Anosov groups are Hausdorff measures

Group Actions, Geometry and Dynamics
Event time: 
Monday, February 5, 2024 - 4:00pm
Subhadip Dey
Speaker affiliation: 
Yale University
Event description: 
In the theory of Kleinian groups, Sullivan’s classical theorem establishes the correspondence between Patterson-Sullivan measures and Hausdorff measures on the limit sets for convex cocompact Kleinian groups. This connection provides a geometric understanding of Patterson-Sullivan measures, emphasizing their association with the internal metric on limit sets. Recent advancements in the theory of infinite co-volume discrete subgroups of higher-rank Lie groups have brought Anosov subgroups into focus as a natural higher-rank extension of convex cocompact Kleinian groups. This raises an intriguing question: under what conditions do Patterson-Sullivan measures for Anosov subgroups emerge as Hausdorff measures on limit sets with appropriate metrics?  In this talk, we disuss  joint work with Dongryul Kim and Hee Oh, which provides a definitive answer to this question. We will also discuss several applications, including  the analyticity of (p,q)-Hausdorff dimensions as functions on the Teichmuller spaces and spectral properties of the associated locally symmetric manifolds.