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.
Proves methods for creating convex projective 3-manifolds with cusps.
problem Creating convex projective 3-manifolds with generalized cusps.
method Properly convex deformations of hyperbolic structures, controlling cusp types.
result First known example of a 1-cusped hyperbolic 3-manifold with a type 2 cusp.
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.
Generalizes DW invariants for cusped 3-manifolds, distinguishing some pairs.
problem Distinguishing cusped 3-manifolds with same volumes and invariants.
method Introducing and analyzing generalized Dijkgraaf-Witten invariants.
result Generalized DW invariants can distinguish some pairs of cusped hyperbolic 3-manifolds.
4-manifolds show every flat 3-manifold as cusp sections.
problem Realizing flat 3-manifolds as cusp sections of hyperbolic 4-manifolds.
method Transitive action on cusps, dense flat metrics realization.
result Existence of many cusp-transitive 4-manifolds.
Study of isometries on hyperbolic 3-manifold cusps.
problem Understanding transitivity in hyperbolic 3-manifold actions.
method Analyzing multiply transitive actions of isometries on cusps.
result Proved a conjecture about the maximum transitivity and upper bounds on cusps.
4 flat 3-manifolds realized in hyperbolic 4-space.
problem Realizing flat 3-manifolds in hyperbolic 4-space.
method Using finite-volume hyperbolic 4-manifolds with cusp sections.
result 6 closed orientable flat 3-manifolds realized as cusp sections.
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.
This paper studies the waist size of cusps in hyperbolic 3-manifolds, proving unique smallest sizes for specific manifolds.
problem Determining the smallest waist sizes of cusps in hyperbolic 3-manifolds.
method Analyzing the shortest nontrivial curves generated by parabolic isometries in maximal cusp boundaries.
result The next two smallest waist sizes are realized uniquely for specific manifolds.
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.
Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.
problem Finding the minimum volume of a 3-cusped orientable hyperbolic 3-manifold.
method Using guts in sutured and pared manifolds.
result The volume of a 3-cusped orientable hyperbolic 3-manifold is at least 5.49... = 6 × Catalan's constant.
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.
The paper proves an equality involving torsion, volume, and a product for hyperbolic 3-manifolds.
problem Understanding the geometry and topology of hyperbolic 3-manifolds with cusps.
method Using Reidemeister torsion, complex volume, and Zograf's infinite product.
result Proves an equality linking these mathematical concepts.
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.
Cusped hyperbolic 3-manifolds can have up to 4 cusps under certain group actions.
problem Understanding the maximum number of cusps in hyperbolic 3-manifolds under group actions.
method Analyzing the isometry group actions on cusps of hyperbolic 3-manifolds.
result Constructing a family of manifolds with no upper bound on the number of cusps for k=2. New math shows many 3D hyperbolic shapes can fit together.
problem Counting specific 3D shapes that fit together.
method Examined both arithmetic and non-arithmetic shapes, focusing on their volume.
result The number of such shapes grows super-exponentially with volume.
New hyperbolic 3-manifolds found that can't be mirrored.
problem Finding achiral hyperbolic 3-manifolds not homeomorphic to amphicheiral knot complements.
method Examined 1-cusped hyperbolic 3-manifolds and knot complements in closed 3-manifolds.
result Infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to amphicheiral knot complements.
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 of (R,ε)-panted cobordism groups in cusped hyperbolic 3-manifolds.
problem Understanding the structure of (R,ε)-panted cobordism groups in cusped hyperbolic 3-manifolds. method Investigating the abelian group generated by (R,ε)-good curves in M modulo the oriented boundaries of (R,ε)-good pants. result For sufficiently small ε>0 and sufficiently large R>0, the (R,ε)-panted cobordism group of M is isomorphic to H1(extSO(M);Z). 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.
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.
New hyperbolic 3-manifolds with multiple cusps are found that sound the same but look different.
problem Finding hyperbolic 3-manifolds with multiple cusps that are isospectral but not isometric.
method Used Sunada's method and the Strong Approximation Theorem of Nori and Weisfeiler.
result Constructed hyperbolic 3-manifolds with multiple cusps that are isospectral but not isometric.
This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also sho…
The paper explores anomalous subvarieties in hyperbolic 3-manifolds and their geometric implications.
problem Understanding anomalous subvarieties in holonomy varieties of hyperbolic 3-manifolds.
method Analyzing the structure of anomalous subvarieties and their relation to geometric properties of hyperbolic 3-manifolds.
result Maximal anomalous subvarieties of holonomy varieties correspond to specific geometric configurations of cusps in hyperbolic 3-manifolds.
Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, incl…
New findings show some hyperbolic 3-manifolds can't be geometrically bounded.
problem Understanding which cusped hyperbolic 3-manifolds can be geometrically bounded.
method Analyzing embeddings and geometric properties of hyperbolic 3-manifolds.
result Some cusped hyperbolic 3-manifolds cannot be geometrically bounded.
Identifies 3D-index as invariant for cusped hyperbolic 3-manifolds.
problem Invariance of q-series for cusped hyperbolic 3-manifolds.
method Relates Frohman-Kania-Bartoszynska's q-series to Dimofte-Gaiotto-Gukov's 3D-index and tetrahedron index.
result Topological invariance of Frohman-Kania-Bartoszynska's q-series.
Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…
We construct infinitely many examples of pairs of isospectral but non-isometric 1-cusped hyperbolic 3-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are fin…
Profinite rigidity of certain 3-manifolds detected through Dehn fillings.
problem Detecting profinite rigidity in 3-manifolds.
method Observation of profinite isomorphisms between cusped hyperbolic 3-manifolds and their Dehn fillings.
result Some cusped hyperbolic 3-manifolds are profinitely rigid.
3-manifolds with similar completions have matching slopes and polynomials.
problem Matching slopes and polynomials in cusped hyperbolic 3-manifolds.
method Prove similarity of profinite completions leads to matching A-polynomials and boundary slopes. result Strongly detected boundary slopes match between manifolds with similar completions.
We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also nee…
Constructs stable maps from 3-manifolds to surfaces without cusps.
problem Creating stable maps from 3-manifolds to surfaces without problematic points.
method Visual construction of stable maps with specific properties.
result Obtains stable maps with no cusps and specific fiber structures.
Study of hyperbolic 3-manifolds via fractional Dehn twists and cusp geometry.
problem Understanding the geometry of fibred hyperbolic 3-manifolds via combinatorial data.
method Relating Euclidean cusp geometry to fractional Dehn twist coefficients of monodromies.
result Uniform bounds on fractional Dehn twist coefficients for certain open book decompositions.
We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.
Paper finds infinite family of minimal triangulations for complex 3D shapes.
problem Finding minimal ideal triangulations for complex 3D shapes.
method Examined Dehn fillings on specific links to find minimal triangulations.
result Found an infinite family of minimal ideal triangulations for a specific type of 3D shape.
Study on braids and their impact on hyperbolic 3-manifolds' orderability.
problem Investigating the relationship between braids and the orderability of hyperbolic 3-manifolds.
method Examined faithful representations of braid groups on automorphism groups of free groups, focusing on bi-orderability.
result Found that the bi-orderability of hyperbolic 3-manifolds' fundamental groups is linked to the braids used.
Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Regularity of isomorphisms between cusped hyperbolic 3-manifolds.
method Proves regularity of profinite completions of fundamental groups.
result Proves A-polynomial is a profinite invariant. Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.
problem Bounding the L2-norm of harmonic forms in hyperbolic 3-manifolds. method Using Thurston norm and interaction with minimal surfaces.
result Generalizes inequalities of Brock-Dunfield and studies sharpness in closed and cusped cases.
Classifies topological types of unions of 3-punctured spheres in hyperbolic 3-manifolds.
problem Classifying unions of 3-punctured spheres in hyperbolic 3-manifolds.
method Topological classification based on geometric properties of 3-punctured spheres.
result Special types of unions appear in only one or a few hyperbolic 3-manifolds.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.
problem Characterizing and computing maximal cusp areas in hyperbolic 3-manifolds.
method Developed a new tiling algorithm and provided simpler expressions for distances.
result Completely characterized the space of cusp neighborhoods and found the Epstein-Penner decomposition.
In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index…
The paper bounds Pachner moves and systoles in hyperbolic 3-manifolds.
problem Bounding Pachner moves and systoles in cusped hyperbolic 3-manifolds.
method Using geometric ideal triangulations and dihedral angles, the paper gives bounds on Pachner moves and systoles.
result Lower bounds on systole length and Pachner move sequence length.
Power series invariant of hyperbolic 3-manifolds matches knot invariants.
problem Understanding topological invariants of hyperbolic 3-manifolds.
method Perturbative power series associated with ideally triangulated cusped hyperbolic 3-manifolds.
result The power series agrees with Kashaev and Andersen-Kashaev invariants to all orders.
Study bounds on cusp volumes of alternating knots on surfaces.
problem Bounding cusp volumes of knots on surfaces.
method Analyzing hyperbolic knots with alternating projections on embedded surfaces.
result Two-sided bounds on cusp area in terms of twist number and surface genus.
3-manifolds have covers with infinitely many ideal triangulations.
problem Proving the existence of infinitely many geometric ideal triangulations in certain 3-manifolds.
method Using separability of peripheral subgroups and conjugacy separability theorems.
result Every cusped hyperbolic 3-manifold has a cover with infinitely many geometric ideal triangulations.