New noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
Study shows arithmetic properties of specific hyperbolic Dehn fillings.
problem Arithmeticity of one-cusped Dehn fillings of specific link complements.
method Investigation of cusp fields, trace fields, and invariant trace fields.
result No one-cusped hyperbolic Dehn filling of the Berge manifold is arithmetic.
New recursion found for hyperbolic sphere volumes.
problem Volume calculation of hyperbolic sphere moduli spaces.
method Proved a non-linear recursive relation.
result Generalized Zograf's result for conical points and geodesic boundaries.
Estimates for geodesics on hyperbolic tori improve previous bounds.
problem Counting simple closed geodesics on hyperbolic tori.
method McShane-Rivin norm balls and Markoff numbers.
result The number of simple closed geodesics of length exactly L≥2 is at most CX(logL)2. The paper calculates the growth rates of billiard languages in hyperbolic polygons.
problem Computing the exponential growth rates of billiard languages in polygons.
method New methods relating to minimal tiling paths.
result Explicit computation of exponential growth rates for q even, and bounds for q odd. Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.
problem Counting minimal Lagrangians in hyperbolic surfaces.
method Uses Mirzakhani functions to show growth rate and proves rigidity of area spectrum.
result Number of minimal Lagrangians grows as A6(k−1). Extends spinor-horosphere correspondence to higher dimensions and new spinor types.
problem Constructing isomorphisms between spinors and horospheres in hyperbolic spaces.
method Generalizes Mathews' results to Lipschitz spinors and null multiflags in higher-dimensional hyperbolic spaces.
result Equivariant correspondence between two-component Lipschitz spinors and higher-dimensional horospheres.
New theorem shows embedding impossibility for certain 3-manifolds.
problem Embedding restrictions in 3-manifolds.
method Construction of hyperbolic 3-manifolds with complicated ideals.
result Existence of non-embeddable manifolds in certain 3-manifolds.
Study on bending deformations in hyperbolic manifolds, generalizing Johnson and Millson's work.
problem Understanding infinitesimal deformations in branched bending complexes.
method Defining branched bending deformations, giving lower bounds, and constructing examples.
result Lower bounds on the dimension of deformation spaces and examples of specific deformations.
Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.
problem Understanding stable commutator length in 3-manifolds.
method Explicit quasimorphisms for generic case, hyperbolic geometry for exceptional case.
result Explicit uniform gap of 1/36 for all orbifolds except a sphere with three cone points.
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
Anosov flows found on many hyperbolic 3-manifolds.
problem Existence of Anosov flows on hyperbolic 3-manifolds.
method Explicit construction of Anosov flows on fibered hyperbolic 3-manifolds.
result Positive density of fibered hyperbolic manifolds carrying Anosov flows.
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.
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
Uniform linear bounds on volume changes in 3D hyperbolic spaces.
problem Volume variation in hyperbolic 3-manifolds.
method Uniform linear bounds proof for drilling and filling operations.
result Uniform linear bounds on volume variation proved.
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
Paper develops geometry for Kleinian groups using Farey polynomials.
problem Understanding the geometry of Kleinian groups generated by parabolic elements.
method Sakuma-Weeks triangulations and Farey recursive polynomials.
result Simple recursive algorithm to determine link complement geometry.
Brunnian theta curves in 3D spheres have hyperbolic exteriors.
problem Characterizing Brunnian theta curves in 3D spheres.
method Sutured manifold theory, classification of annuli, and Thurston's hyperbolic geometry.
result Brunnian theta curves of low bridge number have hyperbolic exteriors with totally geodesic boundary.
Software finds ideal polyhedra with rational dihedral angles and volume maxima.
problem Finding ideal convex polyhedra with maximal volume in hyperbolic 3-space.
method Rivin's variational characterization and combinatorial optimization algorithms.
result Maximal volume ideal polyhedra have dihedral angles that are rational multiples of π.
A bijection proves a polynomial volume for genus-0 hyperbolic surfaces with boundaries.
problem Proving the Weil-Petersson volume polynomial in boundary lengths for genus-0 surfaces.
method Generalizing a tree bijection to handle geodesic boundaries, extending spine construction.
result Explicit formula for three-point function in Weil-Petersson random surfaces.
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.
Researchers create coordinates for hyperbolic surfaces, proving a magic formula.
problem Constructing coordinates for hyperbolic structures on genus-2 surfaces.
method Developed Fenchel-Nielsen coordinates and Wolpert's magic formula analogues.
result Found Darboux charts for the Goldman symplectic form on branched hyperbolic structures.
Study b-6j symbols linking anti-de Sitter tetrahedra to hyperbolic geometry.
problem Analyzing b-6j symbols for quantum invariants. method Examining asymptotics and analytic extensions of 6j-symbols. result Connection between anti-de Sitter tetrahedra and hyperbolic geometry.
Criteria for loop separability on surfaces using Goldman bracket.
problem Determining when two loops on an oriented surface can have disjoint representatives.
method Algebraic criteria in terms of the Goldman bracket, extended using hyperbolic geometry.
result Explicit algebraic conditions for loop separability on surfaces.
New representations preserve hyperbolicity but not Fuchsian property.
problem Identifying non-Fuchsian, hyperbolic-preserving representations on surfaces.
method Analyzing fundamental group representations and their properties.
result Non-Fuchsian representations exist with specific Euler classes.
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.
Random hyperbolic surfaces have nearly optimal spectral gaps.
problem Proving the nearly optimal spectral gap conjecture for random Belyi surfaces.
method Using the Brooks-Makover model, the authors show a spectral gap greater than 1/4 - c/log(n).
result A random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than 1/4 - c/log(n).
Embeds surfaces in hyperbolic and anti-de Sitter spaces.
problem Embedding quasi-circles in hyperbolic and anti-de Sitter spaces.
method Using conformal metrics with bounded curvature and derivatives, constructing smooth embeddings.
result Smooth embeddings of surfaces can be constructed to match given boundaries.
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.
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.
The study limits the number of ribbon concordant fibered knots.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Combining Floer homology results with fixed points of monodromy and volume inequalities.
result There are only finitely many hyperbolic fibered knots ribbon concordant to any given knot.
New bounds link Schwarzian derivative to hyperbolic geometry.
problem Quantify the relationship between Schwarzian derivative and hyperbolic geometry.
method Established explicit quantitative bounds between Schwarzian norm and bending norm.
result Explicit bounds on bending norm for univalent maps with small Schwarzian norm.
Minimal covolume group found in hyperbolic 3-space.
problem Finding groups with minimal covolume in hyperbolic 3-space.
method Proved existence of a specific group with minimal covolume.
result Minimal covolume group found with covolume equal to Catalan's constant.
Random surfaces have a strong spectral gap with polynomial rate.
problem Understanding spectral gaps in random hyperbolic surfaces.
method Adapting polynomial method for random matrices to Laplacian on surfaces.
result Laplacian spectral gap at least 1/4 - O(1/g^c) for large g.
New invariants from quantum group theory for hyperbolic 3-manifolds.
problem Computing invariants for hyperbolic 3-manifolds with boundary.
method Using modular doubles of quantum sl(2;R) and 6j-symbols. result Invariants decay exponentially with hyperbolic volume and 1-loop terms.