2023/02/22 by Dongryul M. Kim, Yair N. Minsky, Kim, Dongryul M. +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2302.11100
openalex publication_date 2023/02/22 · openalex created_date 2023/02/24 · openalex updated_date 2026/07/28
The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup Γ<SO^∘ (n,1), n≥ 2, the Hausdorff dimension of the limit set of Γ is equal to the critical exponent of Γ. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let Δ be a finitely generated group and ρi:Δ→ SO^∘(ni,1) be a convex cocompact faithful representation of Δ for 1≤ i≤ k. Associated to ρ=(ρ1, ⋯, ρk), we consider the following self-joining subgroup of ∏i=1k SO(ni,1): Γ=(∏i=1kρi)(Δ)=\(ρ1(g), ⋯, ρk(g)):g∈ Δ\ . (1). Denoting by Λ⊂ ∏i=1k \mathbbSni-1 the limit set of Γ, we first prove that dimH Λ=max1≤ i≤ k δρi where δρi is the critical exponent of the subgroup ρi(Δ). (2). Denoting by Λu⊂ Λ the u-directional limit set for each u=(u1, ⋯, uk) in the interior of the limit cone of Γ, we obtain that for k≤ 3, (ψΓ(u))/(maxi ui )≤ dimH Λu ≤ (ψΓ(u))/(mini ui ) where ψΓ:ℝk→ ℝ∪\-∞\ is the growth indicator function of Γ.