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- Title
Torus counting and self-joinings of Kleinian groups.
- Authors
Edwards, Sam; Lee, Minju; Oh, Hee
- 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 풫 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 ∈ 풫 : Vol (T) > s , T ∩ E ≠ ∅ } = c 풫 ⋅ 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.
- Subjects
TORUS; CIRCLE; ASYMPTOTIC distribution; ORBITS (Astronomy); GROUP theory; COUNTING
- Publication
Journal für die Reine und Angewandte Mathematik, 2024, Vol 2024, Issue 807, p151
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2023-0089