A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study spectral gaps and bass notes of random hyperbolic 3-orbifolds.
We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the fall of 2013.
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fun…
The paper connects Apollonian packings to knot theory and improves link representations.
We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…
Study of elementary planes in Apollonian orbifold with unusual equidistribution.
Counting spheres in hyperbolic space with effective methods.
Study generates infinite circle packings with a specific property.
Let H denote the standard one-point completion of a real Hilbert space. Given any non-trivial proper sub-set U of H one may define the so-called `Apollonian' metric d_U on U. When U \subset V \subset H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction p…
Study shows Julia sets and gasket limit sets are quasiconformally different.
Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.
Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the s…
In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.
The paper explores orthogeodesics on hyperbolic surfaces and their integer traces.
Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…
Survey explores interactions between four conformal dynamics branches.
Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
Several authors have pointed out the connection between Barbilian's metric introduced in 1934 and the recent study of Apollonian metrics. We provide examples of various distances that can be obtained by Barbilian's metrization procedure and we discuss the relation between this metrization procedure and important Rieman…
Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family…
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
The study proves super-rigidity of Gromov's random monster group for various types of groups.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Virtual twin groups map to symmetric groups, revealing automorphism structure.
Characterizes group connections on group bundles.
Study on totally symmetric sets with group applications.
Affine cactus groups are CAT(0) and hyperbolic.
The study restricts groups in graph of groups structures.
New Garside structures found for torus knot groups and related braid groups.
Logarithmic separation profile in hyperbolic groups shows hierarchical structure.
New Garside structures derived from groups, leading to new group properties.
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…
The group of 2-by-2 matrices with integer entries and determinant can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, name…
The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
New reflection groups derived from torus knots with finite meridians.
Graphically discrete groups have strong rigidity properties.
Paper proves vanishing homology groups for certain hyperbolic groups.
Study fundamental groups of geometric transformation groups using loop spaces.
Simple construction of Lie 2-groups from loop group extensions.
Study knot invariants using automorphism groups of free nilpotent groups.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…
The study shows that certain groups can be uniquely identified by their finite abelian summands.
Survey on Coxeter groups for Lie group examples.
In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of …
The paper studies actions on Bass-Serre trees and identifies new -simple groups.
In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself throug…