Torus counting and self-joinings of Kleinian groups

2022/10/19 by S. F. Edwards, Edwards, Sam, Minju Lee +3 · 1 citation
Mathematics · #37A17 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2210.10229

openalex publication_date 2022/10/19 · openalex created_date 2022/10/22 · openalex updated_date 2026/07/28

Abstract

For any d≥ 1, we obtain counting and equidistribution results for tori with small volume for a class of d-dimensional torus packings, invariant under a self-joining Γ_ρ<∏_i=1^dPSL₂(ℂ) of a Kleinian group Γ formed by a d-tuple of convex cocompact representations ρ=(ρ₁, ⋯, ρ_d). More precisely, if \mathcal P is a Γ_ρ-admissible d-dimensional torus packing, then for any bounded subset E⊂ ℂ^d with ∂ E contained in a proper real algebraic subvariety, we have lim_{s→ 0} s^δ_L¹(ρ) ⋅ #\T∈ P: Vol (T)gt; s, T∩ E≠ ∅ \= c_{\mathcal P}⋅ ω_ρ (E∩ Λ_ρ). Here 0<δ_L¹(ρ)≤ 2/√ d is the critical exponent of Γ_ρ with respect to the L¹-metric on the product ∏_i=1^d ℍ³, Λ_ρ⊂ (ℂ∪\∞\)^d is the limit set of Γ_ρ, and ω_ρ is a locally finite Borel measure on ℂ^d∩ Λ_ρ which can be explicitly described. The class of admissible torus packings we consider arises naturally from the Teichmüller theory of Kleinian groups. Our work extends previous results of Oh-Shah on circle packings (i.e. one-dimensional torus packings) to d-torus packings.

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