Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Paper constructs infinitely many non-braid positive hyperbolic L-space knots.
problem Proving the non-braid positivity of hyperbolic L-space knots.
method Constructs infinitely many hyperbolic L-space knots that are not braid positive.
result Distinct examples from Baker and Kegel's constructions.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
Hyperbolic groups' infinite orbits spread evenly in spaces.
problem Equidistribution of hyperbolic groups in homogeneous spaces.
method Averaging measures along spheres in Cayley graphs converges to Haar measure.
result Infinite orbits of hyperbolic groups equidistribute in homogeneous spaces.
Study of infinite type surfaces' mapping class groups via hyperbolic structures.
problem Understanding mapping class groups of infinite type surfaces.
method Definition of a topology on a moduli space of marked hyperbolic structures.
result Continuous action of mapping class groups on the marked moduli space.
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
Paper finds first infinite family of hyperbolic knots with specific properties.
problem Identifying new hyperbolic knots with specific properties.
method Examined SnapPy census and used knot theory to find new infinite family.
result First infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two.
The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.
problem Characterize infinite circle patterns and convex polyhedra in hyperbolic 3-space.
method Extends techniques from previous work to prove rigidity and uniformization theorems for infinite circle patterns and convex polyhedra.
result Establishes existence and rigidity of infinite regular circle patterns and convex trivalent polyhedra.
We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…
New families of hyperbolic polyhedra yield infinitely many unique reflection groups.
problem Understanding commensurability classes of compact Coxeter polyhedra in hyperbolic spaces.
method Analyzing families of compact Coxeter polyhedra constructed by Makarov.
result Proves infinitely many commensurability classes in 4- and 5-dimensional hyperbolic spaces.
New groups found in hyperbolic space with infinite fields of definition.
problem Finding maximal reflection groups in hyperbolic space.
method Developed new quasi-arithmetic reflection groups for hyperbolic 2-space.
result Infinitely many maximal quasi-arithmetic reflection groups with unbounded field degrees.
A 3D space of hyperbolic manifolds is connected but not path-connected.
problem Proving connectivity and non-path-connectedness of framed hyperbolic 3-manifolds.
method Two proofs using density theorems for Kleinian groups, constructing dense sets of framings, and discussing paths.
result The space of framed infinite volume hyperbolic 3-manifolds is not path-connected.
New infinite family of knots found with unique properties.
problem Finding asymmetric L-space knots with specific properties.
method Generalizing known asymmetric L-space knot t12533 to form an infinite family.
result First infinite family of asymmetric hyperbolic L-space knots of braid index 4.
Let X0 be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric …
Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we i…
3D hyperbolic spaces have endless simple paths.
problem Finding simple paths in complex 3D spaces.
method Analyzing geodesics in hyperbolic 3-manifolds.
result Cusped hyperbolic 3-manifolds have infinitely many simple closed geodesics.
Study classifies mapping class groups with hyperbolic actions on infinite-type surfaces.
problem Classifying mapping class groups with hyperbolic actions on infinite-type surfaces.
method Analyzing curve graphs and their translates to construct hyperbolic spaces.
result Classification of mapping class groups with hyperbolic actions.
Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Study ideal circle patterns (ICPs) and develop a uniform Ring Lemma via pointed Gromov-Hausdorff convergence.
result Establish existence and rigidity of embedded ICPs and infinite ideal polyhedra (IIP).
Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topological infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, …
New representation of PSL2(R) on infinite hyperbolic space via convex bodies.
problem Continuous irreducible actions of PSL2(R) on infinite-dimensional hyperbolic space.
method Using hyperbolic model for convex bodies, produce a continuous and irreducible representation.
result Yields a convex cocompact PSL2(R)-action on infinite-dimensional hyperbolic space with specific quotient properties.
The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface Sm,s with s ends and exactly m of them with infinite genus, such that m,s∈N and 1<m≤s, we give a precise description of t…
Graph products inherit Morse local-to-global property from their components.
problem Generalizing local-to-global property to graph products of infinite groups.
method Generalizing maximization procedure for relatively hierarchically hyperbolic groups and showing stable embeddings.
result Graph products of infinite Morse local-to-global groups have the Morse local-to-global property.
We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total spaces of orientable vector bundles over M.
Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.
problem Characterize infinite circle patterns in the Weil-Petersson class.
method Investigate circle patterns parameterized by discrete harmonic functions of finite Dirichlet energy, equipped with a Riemannian metric.
result Induced quasiconformal homeomorphisms from the unit disk to itself belong to the Weil-Petersson class.
In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half…
Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
problem Characterizing the hyperbolicity of automorphism groups of infinitely-ended groups.
method Proving acylindrical hyperbolicity through group properties and automorphism analysis.
result Automorphism groups of infinitely-ended groups are acylindrically hyperbolic.
A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient…
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
This paper explores infinite-dimensional Teichmüller spaces and their properties.
problem Teichmüller spaces of infinite-type surfaces are complex and depend on base structures.
method Study various distance functions and Teichmüller spaces associated with infinite-type surfaces.
result Finitely supported Teichmüller space is dense in asymptotically isometric Teichmüller space.
Study complex reflections in infinite Coxeter tetrahedron moduli space.
problem Characterize representations of Coxeter group in complex hyperbolic space.
method Type-preserving representations of Coxeter group G to PU(3,1), parameterized by θ. result Discrete and faithful representations for θ∈[65π,π]. First nontrivial moduli space in complex hyperbolic space. In this paper we study the homogeneous Kaehler manifolds (h.K.m.) which can be Kaehler immersed into finite or infinite dimensional complex space forms. On one hand we completely classify the h.K.m. which can be Kaehler immersed into a finite or infinite dimensional complex Euclidean or hyperbolic space. Moreover, we e…
Classifies tight contact structures on surgeries of the Whitehead link.
problem Classifying tight contact structures on surgeries of the Whitehead link.
method Analyzes various surgeries on the Whitehead link to classify tight contact structures.
result Determines tight contact structures, Stein fillability, and virtually overtwisted properties.
Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.
problem Finding isotopic links in 3-sphere with specific properties.
method Using double covers and Reidemeister moves to construct and analyze links.
result Infinitely many non-isotopic hyperbolic links in lens space have isotopic lifts in 3-sphere.
Uniform undistortion in cyclic subgroups of certain groups.
problem Understanding undistorted subgroups in group actions.
method Using quasimorphisms and hierarchically hyperbolic groups.
result Sharp examples of undistorted subgroups in hierarchically hyperbolic groups.
The paper solves curvature problems on infinite hyperbolic surfaces.
problem Prescribed curvature flow on hyperbolic surfaces with infinite topological type.
method Introduced a prescribed curvature flow adapted to infinite cellular decompositions.
result Established well-posedness of the flow and proved convergence under certain conditions.
Classifies big mapping classes on infinite type surfaces.
problem Investigating geometric properties of hyperbolic structures on infinite type surfaces.
method Analyzing the space of convex hyperbolic structures with Fenchel-Nielsen topology and studying the dynamics of mapping classes.
result Classifies big mapping classes into three types based on their dynamics on Teichmüller subspaces.
By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…
Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of Kähler groups on infinite dimensional re…
The study proves limitations on isospectral hyperbolic surfaces with discrete length spectra.
problem Characterizing isospectral hyperbolic surfaces with discrete length spectra.
method Utilizing Sunada's method and topological self-duplicating ends, the study explores isospectral families and their cardinality.
result Finite groups can be realized as full isometry groups of hyperbolic structures with discrete spectrum on surfaces with self-duplicating ends.
Study shows infinite hyperbolic knots with odd torsion in Khovanov homology.
problem Understanding torsion in Khovanov homology for hyperbolic knots.
method Analyzes knots in S3 with specific torsion in Khovanov homology. result Infinitely many hyperbolic knots with odd torsion in Khovanov homology.
We show that the verbal width is infinite for acylindrically hyperbolic groups, which include hyperbolic groups, mapping class groups and Out(Fn).
New simplicial complex for infinite-type surfaces shows graph properties.
problem Characterizing infinite-type surfaces using graph theory.
method Constructing grand arc graph and analyzing its properties.
result Grand arc graph is infinite-diameter and δ-hyperbolic under certain conditions.
On the one hand, we construct a continuous family of non-isometric proper CAT(-1) spaces on which the isometry group Isom(Hn) of the real hyperbolic n-space acts minimally and cocompactly. This provides the first examples of non-standard CAT(0) model spaces for simple Lie groups. On the other hand…
Paper introduces combinatorial Ricci flows on infinite disk triangulations.
problem Discrete Ricci flows on infinite triangulations of the open disk.
method Combinatorial Ricci flows (CRFs) in Euclidean and hyperbolic geometries.
result Existence and uniqueness of CRFs, convergence results, uniformization theorem.
We prove that a closed arithmetic hyperbolic 3-manifold with positive first betti number has virtually infinite first betti number.
Infinite hyperbolic knots yield unusual surgeries.
problem Finding knots with specific surgery results.
method Examined infinite families of hyperbolic knots and their surgeries.
result Discovered knots with surgeries producing graph manifolds with five disjoint, non-parallel incompressible tori.