Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
problem Understanding the generic properties of hyperbolic 3-manifolds.
method Counting model on links and Dehn surgeries.
result Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
Study on length spectrum of random hyperbolic 3-manifolds.
problem Understanding the length spectrum of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra and analyzing their length spectrum as volume tends to infinity.
result The length spectrum converges in distribution to a Poisson point process with a computable intensity λ as volume increases.
Study on systole of random hyperbolic 3-manifolds, proving limit exists and calculating it.
problem Understanding the systole of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra, calculating expected systole limit as volume increases.
result Closed formula and numerical approximation for the limit of the expected systole as volume tends to infinity.
Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results…
We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random wal…
A random group contains many subgroups which are isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. These subgroups can be taken to be quasi-isometrically embedded. This is true both in the few relators model, and the density model of random groups (at any density les…
We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.
The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
problem Proving nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
method Using quasi-Fuchsian surfaces and totally geodesic surfaces, the study proves filling properties with rigorous mathematical proofs.
result Proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.
The paper examines random walks on metric spaces and finds commensurable subgroups.
problem Determining commensurable subgroups via stationary measures in metric spaces.
method Analyzing random walks on isometry groups of metric spaces with non-singular stationary measures.
result Subgroups generated by random walks are commensurable under mild conditions.
A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.
The Cannon-Thurston map's measures become singular with respect to sphere measures.
problem Characterizing the behavior of measures under the Cannon-Thurston map.
method Analyzing the geometric properties of hyperbolic geodesics and quasi-geodesics.
result Natural measures on the circle become singular with respect to measures on the sphere.
New model for links uses meander diagrams and combinatorics.
problem Modeling and analyzing random links.
method Random meander model based on meander diagrams and graphs, proving properties using combinatorics.
result Trivial links are unlikely, and there's a lower bound on non-isotopic knots.
The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.
problem Understanding the behavior of geodesics and measures on fibered hyperbolic 3-manifolds.
method Properties of geodesics and measures on the circle and sphere are analyzed to prove singularity.
result Natural measures on the circle become singular with respect to measures on the sphere.
We study the smallest positive eigenvalue λ1(M) of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface of genus g≥2. We show the existence of a constant C>0 only depending on g so that $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vo…
Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume …
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct bou…
We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative C1+ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the co…
New examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.
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.
A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-manifolds which do not satisfy this condition.
Surveying recent progress on hyperbolic 3-manifold rigidity.
problem Profinite rigidity of hyperbolic 3-manifolds.
method Review of profinite completion and rigidity of groups, evidence from other types of 3-manifolds, and existing ideas.
result Positive evidence for profinite rigidity in hyperbolic 3-manifolds.
The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.
problem Understanding subgroups of bounded rank in hyperbolic 3-manifold groups.
method Proving a finiteness theorem for subgroups of bounded rank.
result Every bounded rank covering tower of closed hyperbolic 3-manifolds is a tower of finite covers associated to a fibration over a 1-orbifold.
We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions appears only in a single hyperbolic 3-manifold or Dehn fillings of a single hyp…
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.
problem Proving the existence of hyperbolic structures on 3-manifolds with cusps.
method Combinatorial Ricci curvature flow methods to study pseudo 3-manifolds and ideal triangulations.
result The extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a zero Ricci curvature metric.
3-manifolds are CR uniformized on spheres, proving a conjecture.
problem Uniformizing 3-manifolds with cusps using CR methods.
method Spherical CR uniformization of complex hyperbolic triangle groups.
result Magic 3-manifolds and other cusped 3-manifolds are CR uniformizable.
Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
Census of 10-tetrahedra hyperbolic 3-manifolds with 150,730 new examples.
problem Census of orientable cusped hyperbolic 3-manifolds up to 10 tetrahedra.
method Complete enumeration of minimal ideal triangulations.
result 439,898 new exceptional Dehn fillings and simplest knot exteriors.
The study finds a bound on the shortest geodesics in hyperbolic 3-manifolds.
problem Finding bounds on the shortest geodesics in hyperbolic 3-manifolds.
method Establishing an upper bound for the length of the nth shortest closed geodesic in terms of the volume of the manifold. result An upper bound for the length of the nth shortest closed geodesic in terms of the volume of the manifold. Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Understanding the regularity of isomorphisms between hyperbolic 3-manifolds.
method Strengthened a previous result by proving regularity in the sense of Boileau and Friedl.
result Profinite isomorphisms of hyperbolic 3-manifolds are regular.
Randomly glued tetrahedra form connected 3-manifolds with a single boundary.
problem Understanding the properties of random three-manifolds formed by truncated tetrahedra.
method Asymptotic analysis of random glued manifolds, proving laws of large numbers, and bounding various topological and geometric properties.
result The random manifolds are connected, have a single boundary component, and admit a unique hyperbolic metric with a uniform spectral gap.
The paper finds hyperbolic small knots in many 3-manifolds.
problem Finding small knots in 3-manifolds.
method Explicit examples of hyperbolic small knots in spherical 3-manifolds.
result Explicit examples of hyperbolic small knots in most spherical 3-manifolds.
New bounds found for minimal surfaces in hyperbolic 3-manifolds.
problem Bounding the maximum principal curvatures of minimal surfaces in hyperbolic 3-manifolds.
method Short argument using work of Uhlenbeck and families of fibered hyperbolic 3-manifolds.
result Uniform lower bound for maximum principal curvatures greater than one.
Improved bounds linking entropy and volume in hyperbolic 3-manifolds.
problem Establishing bounds between entropy and volume in hyperbolic 3-manifolds.
method Heegaard Floer homology and hyperbolic geometry.
result Entropy is bounded by hyperbolic volume with logarithmic factor.
We construct a locally hyperbolic 3-manifold M such that π1(M) has no divisible subgroups. We then show that M is not homotopy equivalent to any complete hyperbolic manifold.
Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.
problem Understanding dynamics in hyperbolic 3-manifolds and Seifert manifolds.
method Classification of partially hyperbolic diffeomorphisms and pseudo-Anosov dynamics.
result Complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and Seifert manifolds.
Algorithm decides if two hyperbolic 3-manifolds are homeomorphic.
problem Determining if two hyperbolic 3-manifolds are homeomorphic.
method Algorithm finds hyperbolic structures and compares geometric manifolds.
result Algorithm runs in bounded time for triangulations with at most t tetrahedra. Study on Cheeger constant in specific hyperbolic 3-manifolds.
problem Analyzing Cheeger constant in convex co-compact hyperbolic 3-manifolds.
method Examining Cheeger constant as a functional in the space of specific hyperbolic 3-manifolds.
result Global maximum of Cheeger constant is uniquely attained at the Fuchsian locus.
Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.
problem Characterizing manifolds at infinity of complex hyperbolic orbifolds.
method Spherical CR uniformization and Dehn surgery.
result Specific 3-manifolds at infinity of complex hyperbolic triangle groups are identified.
We count the number of conjugacy classes of maximal, genus g, surface subroups in hyperbolic 3-manifold groups. For any closed hyperbolic 3-manifold, we show that there is an upper bound on this number which grows factorially with g. We also give a class of closed hyperbolic 3-manifolds for which there is a lower bound…
The paper finds lower bounds on hyperbolic 3-manifold volumes.
problem Finding lower bounds on hyperbolic 3-manifold volumes.
method Decomposing a 3-manifold into hyperbolic pieces and summing their volumes.
result The volume of a 3-manifold is bounded below by the sum of the volumes of its hyperbolic pieces.
Infinite family of hyperbolic 3-manifolds with large volumes.
problem Finding large volume 3-manifolds.
method 2-fold branched covers of 3-manifolds M.
result Existence of hyperbolic 3-manifolds with arbitrarily large volume.
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.
Proves unique hyperbolic metric for 3-manifolds with ideal triangulation.
problem Proving a unique hyperbolic metric for 3-manifolds with specific triangulations.
method Combining combinatorial Ricci flow with ideal triangulation for pseudo 3-manifolds.
result Extended Ricci flow converges to the hyperbolic metric exponentially fast.
Proves hyperbolic 3-manifolds have angle structures under certain conditions.
problem Proving angle structures for non-compact hyperbolic 3-manifolds.
method Subdividing ideal polyhedral decompositions and applying topological conditions.
result Proves existence of ideal triangulations with angle structures.