Dr Samuel Edwards samuel.c.edwards@durham.ac.uk
Assistant Professor
Torus counting and self-joinings of Kleinian groups
Edwards, Samuel; Lee, Minju; Oh, Hee
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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