2023/02/07 by Dongryul M. Kim, Hee Oh, Kim, Dongryul M. +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2302.03552
openalex publication_date 2023/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n, m≥ 2. Let Γ<SO^∘(n+1,1) be a Zariski dense convex cocompact subgroup and Λ⊂\mathbbSn be its limit set. Let ρ: Γ→ SO^∘(m+1,1) be a Zariski dense convex cocompact faithful representation and f:Λ→ \mathbbSm the ρ-boundary map. Let Λf:= \bigcup \ C ∩ Λ: \beginmatrix C ⊂ \mathbbSn is a circle such that
f(C ∩ Λ) is contained in a proper sphere in \mathbbSm \endmatrix \. When there exists at least one Λ-doubly stable circle in \mathbbSn (e.g., Ω=\mathbbSn-Λ is disconnected), we prove the following dichotomy: either Λf= Λ or Hδ(Λf) =0, where Hδ is the Hausdorff measure of dimension δ=dimH Λ. Moreover, in the former case, we have n=m and ρ is a conjugation by a Möbius transformation on \mathbbSn. 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 Γρ=(id × ρ)(Γ) < SO^∘(n+1,1)× SO^∘(m+1,1). We also obtain an analogous theorem for any divergence-type subgroup.