Geometric methods study 3-manifold splittings.
problem Studying Heegaard splittings of 3-manifolds.
method Geometric approaches.
result Recent advances in geometric methods.
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48 results for “geometric 3-manifolds”
In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.
We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.
This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
Classifies Morse boundaries of 3-manifold groups.
problem Classifying Morse boundaries of 3-manifold groups.
method Classifies Morse boundaries into 9 types based on geometric decompositions.
result 9 different homeomorphism types of Morse boundaries.
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.
We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…
Classifying spaces for the family of virtually abelian subgroups of orientable -manifold groupsmath.GT
This paper calculates the geometric dimension for 3-manifold groups up to n=2.
problem Computing the geometric dimension for 3-manifold groups.
method Analyzing families of subgroups containing virtually abelian subgroups.
result Computed the geometric dimension for 3-manifold groups for all n ≥ 2.
Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the …
Constructs knots from 3-manifolds with specified geometric limits.
problem Build knots from 3-manifolds with specific geometric properties.
method Constructs explicit families of knots converging to specified 3-manifolds.
result Constructs knots from geometrically finite 3-manifolds with one end.
Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.
problem Algebraic K-theory of 3-manifold groups.
method Farrell-Jones isomorphism conjecture, models for virtually cyclic subgroups, geometrization theorem.
result Descriptions of Whitehead groups and algebraic K-theory groups in terms of finite subgroups and Nil-groups.
We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows tha…
The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompr…
Geometrically interprets symplectic structure in 3-manifold triangulations.
problem Understanding symplectic structures in 3-manifold triangulations.
method Geometric interpretation and algorithm construction for symplectic basis.
result Algorithm constructs curves forming a symplectic basis.
It is well known that an arbitrary closed orientable -manifold can be realized as the unique boundary of a compact orientable -manifold, that is, any closed orientable -manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic -manifold is geometrically bounding…
In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.
We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.
In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating combinatorial descriptions and geometric properties of hyperbolic 3-manifolds.
We show that the number of isometry classes of cusped hyperbolic -manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.
In this note, we show that there exist cusped hyperbolic -manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic -manifolds, and…
New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.
problem Existence of specific geometric structures on 3-manifolds.
method Generalized geometry and surgery techniques.
result Any closed orientable 3-manifold admits a -generalized complex structure.
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.
Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompo…
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 uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
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.
Highly twisted knots can be geometrically triangulated.
problem Proving geometric triangulations for hyperbolic 3-manifolds.
method Using highly twisted knots and extending Gueritaud and Schleimer's work.
result Sufficiently highly twisted knots admit a geometric triangulation.
This paper solves a problem in 3D geometry by defining a canonical partition for certain manifolds.
problem Building an explicit canonical decomposition for orientable 3-manifolds defined by vector-colourings of 3-polytopes.
method Analysis of results from previous studies on similar problems.
result A complete answer to the problem of decomposing orientable 3-manifolds defined by vector-colourings of 3-polytopes.
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
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 tetrahedra.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…
We extend the concept of renormalized volume for geometrically finite hyperbolic -manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold with geometrically finite limit. This allows us to show that the renormalized volume attains its…
Derives geometrically a description of a 3-manifold's second homotopy group.
problem Describing the second homotopy group of a 3-manifold using kernels of homomorphisms.
method Geometric derivation of Lei and Wu's result.
result Geometric derivation of the second homotopy group description.
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.
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.
We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.
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.
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
In this paper, we define a new algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery along a hyperbolic knot complement in S^3. We establish a Casson type invariant for these 3-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some ap…
We prove a partial generalization of Bonahon's tameness result to surfaces inside irreducible 3-manifolds with hyperbolic fundamental group. Bonahon's result states that geometrically infinite ends of freely indecomposable hyperbolic 3-manifolds are simply degenerate. It is easy to see that a geometrically infinite end…
New findings link 3D shapes to group properties.
problem Understanding groups with specific geometric properties.
method Analyzing spherical Plateau problems and 3-manifolds.
result Isometric solutions to Plateau problems imply geometric properties of groups.
Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…
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…
Classifies horocycle flow closures in hyperbolic 3-manifolds.
problem Classifying horocycle flow closures in hyperbolic 3-manifolds.
method Classifies orbit closures of the 1-dimensional horocycle flow on the frame bundle of M.
result The closure of a horocycle in M is a properly immersed submanifold.
The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.
problem Classifying group-actions on surfaces of small genus, particularly focusing on bounding and geometrically bounding cases.
method Analyzing large group-actions on surfaces of genus 3, distinguishing between bounding and geometrically bounding cases.
result Identifies which large group-actions on surfaces of genus 3 are bounding or geometrically bounding.