Torus counting and self-joinings of Kleinian groups

Abstract

Abstract For any integer d ≥ 1 {d\geq 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 d PSL 2 ⁡ ( ℂ ) {\Gamma_{\rho}<\prod_{i=1}^{d}\operatorname{PSL}_{2}(\mathbb{C})} of a Kleinian group Γ formed by a d-tuple of convex-cocompact representations ρ = ( ρ 1 , … , ρ d ) {\rho=(\rho_{1},\dots,\rho_{d})} . More precisely, if 𝒫 {\mathcal{P}} is a Γ ρ {\Gamma_{\rho}} -admissible d-dimensional torus packing, then for any bounded subset E ⊂ ℂ d {E\subset\mathbb{C}^{d}} with ∂ ⁡ E {\partial E} contained in a proper real algebraic subvariety, we have lim s → 0 ⁡ s δ L 1 ⁢ ( ρ ) ⋅ # ⁢ { T ∈ 𝒫 : Vol ⁡ ( T ) > s , T ∩ E ≠ ∅ } = c 𝒫 ⋅ ω ρ ⁢ ( E ∩ Λ ρ ) . \lim_{s\to 0}{s^{\delta_{L^{1}}({\rho})}}\cdot\#\{T\in\mathcal{P}:% \operatorname{Vol}(T)>s,\,T\cap E\neq\emptyset\}=c_{\mathcal{P}}\cdot\omega_{% \rho}(E\cap\Lambda_{\rho}). Here δ L 1 ⁢ ( ρ ) {\delta_{L^{1}}(\rho)} , 0 < δ L 1 ⁢ ( ρ ) ≤ 2 / d {0<\delta_{L^{1}}(\rho)\leq 2/\!{\sqrt{d}}} , denotes the critical exponent of the self-joining Γ ρ {\Gamma_{\rho}} with respect to the L 1 {L^{1}} -metric on the product ∏ i = 1 d ℍ 3 {\prod_{i=1}^{d}\mathbb{H}^{3}} , Λ ρ ⊂ ( ℂ ∪ { ∞ } ) d {\Lambda_{\rho}\subset(\mathbb{C}\cup\{\infty\})^{d}} is the limit set of Γ ρ {\Gamma_{\rho}} , and ω ρ {\omega_{\rho}} is a locally finite Borel measure on ℂ d ∩ Λ ρ {\mathbb{C}^{d}\cap\Lambda_{\rho}} 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 [H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math. 187 2012, 1, 1–35] on circle packings (i.e., one-dimensional torus packings) to d-torus packings.