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Torus counting and self-joinings of Kleinian groups

Edwards, Samuel; Lee, Minju; Oh, Hee

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Authors

Minju Lee

Hee Oh



Abstract

For any integer 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 d PSL 2 ⁡ ( ℂ ) of a Kleinian group Γ formed by a d-tuple of convex-cocompact representations ρ = ( ρ 1 , … , ρ d ) . More precisely, if 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 L 1 ⁢ ( ρ ) ⋅ # ⁢ { T ∈ 𝒫 : Vol ⁡ ( T ) > s , T ∩ E ≠ ∅ } = c 𝒫 ⋅ ω ρ ⁢ ( E ∩ Λ ρ ) . Here δ L 1 ⁢ ( ρ ) , 0 < δ L 1 ⁢ ( ρ ) ≤ 2 / d , denotes the critical exponent of the self-joining Γ ρ with respect to the L 1 -metric on the product ∏ i = 1 d ℍ 3 , 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 [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.

Citation

Edwards, S., Lee, M., & Oh, H. (2024). Torus counting and self-joinings of Kleinian groups. Journal für die reine und angewandte Mathematik, 2024(807), 151-185. https://doi.org/10.1515/crelle-2023-0089

Journal Article Type Article
Acceptance Date Nov 14, 2023
Online Publication Date Jan 2, 2024
Publication Date Feb 1, 2024
Deposit Date Nov 17, 2023
Publicly Available Date Jan 3, 2024
Journal Journal für die reine und angewandte Mathematik
Print ISSN 0075-4102
Electronic ISSN 1435-5345
Publisher De Gruyter
Peer Reviewed Peer Reviewed
Volume 2024
Issue 807
Pages 151-185
DOI https://doi.org/10.1515/crelle-2023-0089
Public URL https://durham-repository.worktribe.com/output/1931629

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