The paper introduces new invariants to detect hyperbolic parts in relatively hyperbolic groups.
problem Detecting hyperbolic parts in relatively hyperbolic groups.
method Quasi-isometry invariants based on small cancellation theory over free products.
result Infinitely many quasi-isometry types of one-ended hyperbolic relative groups can be constructed.
Book introduces hyperbolic geometry for knot theory.
problem Understanding knots through hyperbolic geometry.
method Explains hyperbolic geometry, geometric structures, and techniques.
result Develops three knot invariants from hyperbolic geometry.
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we pr…
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second …
Study of transitivity in partially hyperbolic maps with expanding linear part.
problem Transitivity of partially hyperbolic endomorphisms with expanding linear part.
method Use of Blichfedt's theorem to analyze dynamical information from homology action.
result Robust transitivity condition and complete dichotomy for special cases.
Introduces new hyperbolicity concepts for complex manifolds.
problem Generalizing hyperbolicity concepts to non-Kähler manifolds.
method Introduces balanced hyperbolicity and divisorial hyperbolicity.
result Every balanced hyperbolic manifold is also divisorially hyperbolic.
Knots with large volume in their thick part are found.
problem Finding hyperbolic knots with significant volume in their thick part.
method Constructing knots in the 3-sphere with specific properties related to injectivity radius.
result There exist knots whose R-thick part volume ratio to total volume is at least 1-ε for any ε > 0.
For large genus, spectral gaps on hyperbolic surfaces approach a limit.
problem Understanding spectral gaps on hyperbolic surfaces of large genus.
method Analyzing the maximum of λk−λk−1 over thick parts of moduli spaces. result The maximum of λk−λk−1 approaches 41 for large genus. This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.
problem Establishing properties of uniformly hyperbolic sets and constructing Markov partitions.
method Backward graph transform, spectral decomposition, shadowing lemma, Markov partitions construction.
result Explicit bounds and Hölder continuity for the coding map.
New definitions and properties for hyperbolic group representations.
problem Understanding representations of hyperbolic groups into affine groups.
method Defining affine Anosov representations and proving their equivalence to Anosov linear parts with proper action.
result Affine Anosov representations of hyperbolic groups have specific properties related to their linear parts and proper actions.
Explains what hierarchically hyperbolic spaces are.
problem None explicitly stated; focuses on definition and understanding.
method Heuristic discussion and detailed technical discussion.
result Provides a mental picture and technical definition of HHSs.
Bounds on homology of hyperbolic orbifolds using simplicial models.
problem Bounding homology of hyperbolic orbifolds.
method Efficient simplicial model of the thick part of hyperbolic orbifolds.
result Linear bounds on Betti numbers and torsion homology in terms of volume.
Study partially hyperbolic diffeomorphisms in 3D, focusing on foliations and dynamics.
problem Classify 3D partially hyperbolic diffeomorphisms homotopic to the identity.
method Analyze Burago and Ivanov's branching foliations in Seifert fibered and hyperbolic manifolds.
result Complete classification of diffeomorphisms in Seifert fibered manifolds, and new potential class in hyperbolic manifolds.
Example of spacetime with causal bubbling, splitting into timelike and spacelike parts.
problem Understanding causal bubbling in spacetimes.
method Example of a globally hyperbolic spacetime with a continuous metric, splitting orthogonally into timelike and spacelike parts.
result The synthetic timelike curvature-dimension (TCD) condition does not prevent causal bubbling.
Effective drilling and filling bounds for hyperbolic 3-manifolds.
problem Understanding changes in metrics and geodesics during Dehn fillings of hyperbolic 3-manifolds.
method Combining tools from Kleinian group theory to transfer results from finite-volume to infinite-volume manifolds.
result Effective bilipschitz and complex length bounds quantifying filling theorems.
Given a closed Riemannian manifold of dimension n and a Morse-Smale function, there are finitely many n-part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of n-part broken trajectories is always at least the hyperbolic volume. The proof…
Study horospheres in hyperbolic 3-manifolds with degenerate ends.
problem Understanding geodesics in degenerate hyperbolic 3-manifolds.
method Analyze almost minimizing geodesics to explore horospheres.
result Identify geodesics passing through thin parts of the manifold.
Study large N oscillations in 3D theories related to black hole physics.
problem Understanding large N sign oscillations in 3D theories via holography.
method Holographic computation of on-shell actions for Euclidean supergravity solutions, Wick rotation of magnetically charged AdS4 black holes.
result Proposed a non-trivial mathematical conjecture regarding phase factors of twisted Reidemeister-Ray-Singer torsion.
We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick pa…
Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on S×R is the tangent bundle of the Teichmüller space of S, if S is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relat…
New formula calculates volumes of ideal hyperbolic drums.
problem Computing volumes of ideal hyperbolic drums.
method Proved a volume formula for arbitrary ideal hyperbolic antiprisms (drums).
result Volume formula for ideal hyperbolic drums.
This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to f…
In this paper, we find infinite hyperbolic 3-manifolds that admit no weakly symplectically fillable contact structures, using tools in Heegaard Floer theory. We also remark that part of these manifolds do admit tight contact structures.
Extends Teichmüller space quasi-isometry to k-multicurve graphs.
problem Quasi-isometric relationship between Teichmüller space and curve graphs.
method Electrifying Teichmüller space along thin part, adapting Lackenby-Yazdi pants graph bounds.
result Quasi-isometry of k-multicurve graph to electrified Teichmüller space. For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…
Survey on spectral gaps of random hyperbolic surfaces.
problem Understanding spectral gaps of random hyperbolic surfaces.
method Brief survey on geometry and spectra, discussion of results by Hide-Magee, Anantharaman-Monk, and Hide-Macera-Thomas.
result Near optimal spectral gaps for random surfaces.
Accessible proof of hyperbolic cubulated groups being virtually special.
problem Proving hyperbolic cubulated groups are virtually special.
method Retained Agol's ideas but made the proof more transparent and detailed.
result Accessible proof of the main theorem.
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.
problem Proving a fundamental gap lower bound for horoconvex domains in hyperbolic space.
method Reduces the problem to a radial-height problem, compares Dirichlet forms with angular operators, and uses Green estimates.
result Establishes a polynomial \(D^{-3}\) scale fundamental gap lower bound.
The paper consists of two parts. In the first one we show that a relatively hyperbolic group G splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…
Crochet models show hyperbolic geometry properties.
problem Visualizing hyperbolic geometry concepts.
method Triangulated hyperbolic plane models.
result Demonstrates various hyperbolic geometry properties.
Study approximates lengths of random hyperbolic surfaces' geodesics.
problem Approximating lengths of geodesics on random hyperbolic surfaces.
method Poisson approximation for the bottom part of the length spectrum.
result Large genus limit of the expected systole computed.
New 3D shapes found without certain flows.
problem Finding 3D shapes without specific flows.
method Using foliations and pseudo-Anosov flows, analyzing cusped hyperbolic 3-manifolds.
result First examples of 3D shapes without veering triangulations.
This is the first in a series of papers showing that Haken manifolds have hyperbolic structures; this first was published, the second two have existed only in preprint form, and later preprints were never completed. This eprint is only an approximation to the published version, which is the definitive form for part I, …
This paper classifies Ricci solitons in complex hyperbolic spaces.
problem Understanding Ricci solitons in complex hyperbolic spaces.
method Analyzing homogeneous expanding Ricci solitons as submanifolds of complex hyperbolic spaces.
result Classification and analysis of Lie subgroups with Ricci soliton induced metric in complex hyperbolic spaces.
Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.
problem Finiteness of arithmetic maximal reflection groups in hyperbolic polyhedra.
method Observation of volume distribution and recent work with M. Fraczyk and S. Hurtado.
result Proof of finiteness of arithmetic maximal reflection groups.
Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... …
We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichmüller space of S, with quasi-convex image lying in the thick part. As a consequence, H^n quasi-isometrically embeds in the curve complex of S.
We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the dimension and the constant used to define the thick-thin decomposition of M.
Introduces Anti-de Sitter geometry and its connection to Teichmüller theory.
problem None explicitly stated, focuses on introducing concepts.
method Broad introduction to Anti-de Sitter geometry, focusing on dimension three and Teichmüller theory.
result Classification of maximal globally hyperbolic Cauchy compact manifolds and construction of the Gauss map.
In a paper with Jean-Paul Dufour in 1999 \cite{DufourZung-Nambu1999}, we gave a classification of linear Nambu structures, and obtained linearization results for Nambu structures with a nondegenerate linear part. There was a case left open in \cite{DufourZung-Nambu1999}, namely the case of smooth linearization of Nambu…
The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal…
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …
New stability theorem for hyperbolic metrics without volume bounds.
problem Stability of finite volume hyperbolic metrics without upper volume bounds.
method Abstract axiomatic framework and bootstrap argument to extend stability result.
result Weaker exponential control of the metric allows for a broader application of the stability theorem.
We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lip…
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.