Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

DOI10.1090/S0894-0347-2011-00691-7zbMath1235.22015OpenAlexW2963318116WikidataQ105978496 ScholiaQ105978496MaRDI QIDQ3011038

Hee Oh, Alex V. Kontorovich

Publication date: 27 June 2011

Published in: Journal of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0894-0347-2011-00691-7

zbMATH Keywords

horospheresKleinian groupsApollonian circle packing

Mathematics Subject Classification ID

Discrete subgroups of Lie groups (22E40) General geometric structures on low-dimensional manifolds (57M50) Circle packings and discrete conformal geometry (52C26)

Related Items (52)

The Apollonian structure of integer superharmonic matrices ⋮ On the rate of equidistribution of expanding horospheres in finite-volume quotients of \(\mathrm{SL}(2,\mathbb{C})\) ⋮ Effective equidistribution of circles in the limit sets of Kleinian groups ⋮ Almost prime Pythagorean triples in thin orbits ⋮ A proof of the positive density conjecture for integer Apollonian circle packings ⋮ Effective circle count for Apollonian packings and closed horospheres ⋮ Classification of joinings for Kleinian groups ⋮ Affine linear sieve, expanders, and sum-product ⋮ Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds ⋮ The sensual Apollonian circle packing ⋮ Skinning measures in negative curvature and equidistribution of equidistant submanifolds ⋮ Integral Apollonian circle packings and prime curvatures ⋮ Hyperbolic systems, zeta functions and other friends ⋮ On realization of isometries for higher rank quadratic lattices over number fields ⋮ On representations of integers in thin subgroups of \(\text{SL}_2(\mathbb Z)\) ⋮ Angles in hyperbolic lattices: The pair correlation density ⋮ Mixing of frame flow for rank one locally symmetric spaces and measure classification ⋮ Arithmetic and dynamics on varieties of Markoff type ⋮ The residual set dimension of a generalized Apollonian packing ⋮ Circle packings from tilings of the plane ⋮ Generalization of Selberg's \(\frac {3}{16} \) theorem and affine sieve ⋮ Group actions and harmonic analysis in number theory. Abstracts from the workshop held May 7--12, 2023 ⋮ Apollonian packings and Kac-Moody root systems ⋮ Torus counting and self-joinings of Kleinian groups ⋮ Semi-hyperbolic rational maps and size of Fatou components ⋮ On the local-global principle for integral Apollonian 3-circle packings ⋮ Strong approximation in the Apollonian group ⋮ Counting common perpendicular arcs in negative curvature ⋮ Distribution of the periodic points of the Farey map ⋮ The asymptotic distribution of circles in the orbits of Kleinian groups ⋮ On the exponential growth of norms in semigroups of linear endomorphisms and the Hausdorff dimension of attractors of projective iterated function systems ⋮ On the local-global conjecture for integral Apollonian gaskets. With an appendix by Péter P. Varjú ⋮ Some Experiments with Integral Apollonian Circle Packings ⋮ From Apollonius to Zaremba: Local-global phenomena in thin orbits ⋮ Counting problems in Apollonian packings ⋮ Equidistribution and counting for orbits of geometrically finite hyperbolic groups ⋮ Quantitative ergodic theorems and their number-theoretic applications ⋮ Apollonian circle packings: dynamics and number theory ⋮ The local-global principle for integral Soddy sphere packings ⋮ On the rate of equidistribution of expanding translates of horospheres in \(\Gamma\backslash G\) ⋮ Apollonian Circle Packings ⋮ Asymptotic Counting in Conformal Dynamical Systems ⋮ A Conjecture on the Hausdorff Dimension of Attractors of Real Self-Projective Iterated Function Systems ⋮ On the limit set of root systems of Coxeter groups acting on Lorentzian spaces ⋮ Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds ⋮ Spatial Statistics of Apollonian Gaskets ⋮ Expander graphs in pure and applied mathematics ⋮ On the deformation of ball packings ⋮ An asymptotic formula for integer points on Markoff-Hurwitz varieties ⋮ A remark on the numerical integration of harmonic functions in the plane ⋮ Filling space with polydisperse spheres in a non-Apollonian way ⋮ Circle packings, kissing reflection groups and critically fixed anti-rational maps

Cites Work

• Unnamed Item
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• Unnamed Item
• Generalization of Selberg's \(\frac {3}{16} \) theorem and affine sieve
• The asymptotic distribution of circles in the orbits of Kleinian groups
• Horocycle flow on geometrically finite surfaces
• Raghunathan's topological conjecture and distributions of unipotent flows
• Apollonian circle packings: Geometry and group theory. III: Higher dimensions
• Affine linear sieve, expanders, and sum-product
• Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices
• Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
• The density at infinity of a discrete group of hyperbolic motions
• The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces
• Invariant measures and minimal sets of horospherical flows
• On Raghunathan's measure conjecture
• The limit set of a Fuchsian group
• Sizing up a solid packing
• Mixing, counting, and equidistribution in Lie groups
• Limit points of Kleinian groups and finite sided fundamental polyhedra
• Apollonian circle packings: Number theory
• Topologie du feuilletage fortement stable. (Topology of the strong stable foliation)
• Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group
• The hyperbolic lattice point count in infinite volume with applications to sieves
• Almost prime Pythagorean triples in thin orbits
• Translate of horospheres and counting problems
• Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
• Congruence Properties of Zariski-Dense Subgroups I
• The Sequence of Radii of the Apollonian Packing
• Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series
• Hausdorff dimension and conformal dynamics, III: Computation of dimension
• Limit sets of discrete groups of isometries of exotic hyperbolic spaces
• Ergodicité et équidistribution en courbure négative
• Equidistribution and counting for orbits of geometrically finite hyperbolic groups
• The Apollonian Packing of Circles
• The Problem of Apollonius
• The Bowl of Integers and the Hexlet
• Hyperbolic manifolds and discrete groups

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