Work

Rigidity of Kleinian groups via self‐joinings: measure theoretic criterion

Year: 2025

Type: article

Abstract: Let . Let be a Zariski dense convex cocompact subgroup and be its limit set. Let be a Zariski dense convex cocompact faithful representation and the ‐boundary map. Let When there exists at least one ‐doubly stable circle in (e.g., is disconnected), we prove the following dichotomy: where is the Hausdorff measure of dimension . Moreover, in the former case, we have and is a conjugation by a Möbius transformation on . Our proof uses ergodic theory for directional diagonal flows and conformal measure theory of discrete subgroups of higher rank semisimple Lie groups, applied to the self‐joining subgroup . We also obtain an analogous theorem for any divergence‐type subgroup.

Source: Journal of Topology

DOI: 10.1112/topo.70035

Authors: Dongryul M. Kim, Hee Oh

Institution: Yale University

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Dynamics and Rigidity for Discrete Subgroups of Lie Groups Beyond Lattices, Dynamics and Kleinian Groups