Study describes prosoluble subgroups in 3-manifold groups.
problem Characterizing prosoluble subgroups in 3-manifold groups.
method Analyzes finitely generated prosoluble subgroups of profinite completions.
result Provides a description of these subgroups.
Extends continuity proof to non-compact manifolds and groups.
problem Automatic continuity for homeomorphism groups of non-compact manifolds.
method Proof extension to non-compact manifolds and groups.
result Homomorphisms from these groups to separable topological groups are continuous.
Study homomorphisms from groups to 3-manifold fundamental groups.
problem Understanding homomorphisms between groups and 3-manifold fundamental groups.
method Proves foundational results and answers specific questions.
result Answers questions posed by Reid-Wang-Zhou and Agol-Liu.
Proves one-relator groups are fundamental groups of Sasakian manifolds.
problem Characterizing groups that can be fundamental groups of Sasakian manifolds.
method Analyzes one-relator groups and classifies groups of deficiency at least two.
result Proves sufficient and necessary conditions for one-relator groups to be fundamental groups of Sasakian manifolds.
Study classifies 2-manifolds with special homeomorphism groups.
problem Classifying 2-manifolds with specific homeomorphism groups.
method Introduced virtual Rokhlin property; classified homeomorphism groups of 2-manifolds.
result 2-manifolds with the virtual Rokhlin property have specific homeomorphism groups.
The paper classifies 4-manifolds based on their fundamental groups and orientation characters.
problem Classifying 4-manifolds with specific fundamental groups and orientation characters.
method Developed a criterion for groups and homomorphisms to classify 4-manifolds up to homotopy equivalence.
result Closed 4-manifolds with specified fundamental groups and orientation characters are classified by their quadratic 2-types.
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
Computes Weyl group of Kähler toric manifold isometries.
problem Computing the Weyl group of Kähler toric manifold isometries.
method Analyzes the group of holomorphic isometries of a Kähler toric manifold with real analytic Kähler metric.
result Computed the Weyl group of the group of holomorphic isometries.
Holonomy groups of K-contact sub-Riemannian manifolds are studied.
problem Understanding the holonomy groups of K-contact sub-Riemannian manifolds.
method Analyzing the horizontal holonomy group and comparing it to the holonomy group of a Riemannian manifold.
result The horizontal holonomy group either coincides with the holonomy group of a Riemannian manifold or is a codimension-one subgroup.
The paper classifies when certain graph braid groups are 3-manifold groups.
problem Identifying when graph braid groups are 3-manifold groups.
method Analyzing the graph braid groups B3(Θm) for specific graphs Θm. result The paper shows that B3(Θ5) is a 3-manifold group, but B3(Θm) is not quasi-isometric to a 3-manifold group for m≥7. The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on b2+ and b2−, the paper constructs these structures. result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.
problem Characterizing and classifying 4-manifolds using Artin presentations and triangle groups.
method Utilizing triangle groups to find Artin presentations that present the trivial group and determining 4-manifolds with specific properties.
result Identified all Artin presentations on two generators that present the trivial group and all smooth, closed, simply-connected 4-manifolds with specific properties.
New manifold structures on Weyl group orbit spaces proven.
problem Constructing generalized Frobenius manifold structures.
method Construction on orbit spaces of affine Weyl groups.
result Monodromy groups are parabolic subgroups.
Proof that specific groups are quasi-isometrically rigid.
problem Quasi-isometric rigidity of specific groups.
method Proof of quasi-isometric rigidity for Kleinian and three-manifold groups.
result Finitely generated Kleinian and three-manifold groups are quasi-isometrically rigid.
We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…
Classifies 4-manifolds with elementary amenable groups and their boundaries.
problem Characterizing compact aspherical 4-manifolds with elementary amenable fundamental groups.
method Classification based on fundamental group properties and Farrell-Jones Conjecture.
result Such manifolds are either polycyclic or solvable Baumslag-Solitar groups.
Free groups can be end homogeneity groups of 3-manifolds.
problem Tackling the possibility of free groups as end homogeneity groups of 3-manifolds.
method Constructing specific 3-manifolds with end homogeneity groups isomorphic to free groups.
result For every finitely generated free group, there exists an irreducible open 3-manifold with that group as its end homogeneity group.
We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) …
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension >2 cannot be a compact Lie group. Hence this holo…
The paper classifies 3-manifold groups with specific torsion elements.
problem Classifying 3-manifold groups with generalized torsion elements of order two.
method Analyzing the fundamental groups of 3-manifolds and their conjugates.
result 3-manifold groups with generalized torsion elements of order two have been classified.
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.
Study properties of group rings of three-manifold groups.
problem Properties of group rings of three-manifold groups.
method By piecing together known facts about three-manifold groups, the paper establishes two properties of the group ring CG. result If G has rational cohomological dimension two, then CG is coherent. If G is torsion-free, then G satisfies the Strong Atiyah Conjecture over C and CG satisfies Kaplansky's Zero Divisor Conjecture. We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, locally symmetric, negatively curved manifolds with cusps. We prove that the gro…
Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.
problem Understanding the structure of diffeomorphism groups of 3-manifolds.
method Deformation retraction and earlier work by multiple authors combined.
result Homotopy type of Diff(X) determined for prime 3-manifolds.
Proves graph 3-manifold groups have two specific properties.
problem Understanding fundamental groups of graph 3-manifolds.
method Constructing sequences of covers to prove properties.
result Graph 3-manifold groups are virtually poly-free and in Lex family.
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
The paper shows how to find inaccessible hyperbolic actions in manifold groups.
problem Finding hyperbolic actions of 3-manifold groups that are inaccessible.
method Analyzing the partial order of cobounded actions on hyperbolic spaces.
result Proves the existence of finite covers with inaccessible fundamental groups.
Classifies manifolds with dense conjugacy classes in their mapping class groups.
problem Classifying manifolds based on conjugacy classes in their mapping class groups.
method Analyzing connected orientable 2-manifolds and their mapping class groups.
result Mapping class groups of certain manifolds have dense conjugacy classes.
We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that all known Ka¨hler groups are Sasaki, in particular, all finite groups are Sasaki. On the other hand, we show there exists many restrictions on the fundamental groups of compact Sasakian manifolds. We als…
This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follow…
We show that if the lower central series of the fundamental group of a closed oriented 3-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion 2-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental group…
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. Einstein metrics are blocked by manifold features and group growth.
problem Existence of Einstein metrics on specific 4-manifolds.
method Analysis of collapsing and group growth effects.
result Several 4-manifolds cannot support Einstein metrics due to specific features.
Study finds specific Lie groups with Kenmotsu structures.
problem Characterizing Lie groups with Kenmotsu structures.
method Determined Lie groups with left invariant Kenmotsu structures.
result These Lie groups are Einstein Riemannian manifolds.
Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.
problem Understanding actions of finite groups on aspherical manifolds.
method Analyzing the outer automorphism group and homeomorphism group of the fundamental group.
result Proves the homeomorphism group is Jordan and bounds the discrete degree of symmetry.
In this work we ask when a group is a 3-manifold group, or more specifically, when does a group presentation come naturally from a Heegaard diagram for a 3-manifold? We will give some conditions for partial answers to this form of the Isomorphism Problem by addressing how the presentation associated to a diagram for a …
The study examines when braid groups of manifolds are Kähler.
problem When are braid groups of manifolds Kähler?
method The approach uses previous results on Kähler groups and homological properties of braid groups.
result With some exceptions, the pure braid group of a Riemann surface with at least 2 strands is never Kähler.
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
Explicit isometry groups found for nearly Kähler manifolds.
problem Understanding the symmetries of nearly Kähler manifolds.
method Alternative, less algebraic approach to find isometry groups.
result Explicit expression for isometry groups of six-dimensional nearly Kähler manifolds.
Automorphism groups of Hopf manifolds are finite and have a bounded order.
problem Understanding the structure of automorphism groups of Hopf manifolds.
method Proved that the automorphism groups of Hopf manifolds are Jordan.
result Automorphism groups of Hopf manifolds are finite and have a bounded order.
Study of Sasaki groups vs Kähler groups, showing distinct behaviors.
problem Comparing fundamental groups of Sasaki and Kähler manifolds.
method Analysis of properties and behavior of Sasaki groups compared to Kähler groups.
result Sasaki groups exhibit distinct behavior from Kähler groups, not closed under direct products, and have upper bounds on dimension.
New examples of knotting phenomena in 4-manifolds with specific fundamental groups.
problem Finding 2-spheres in simply connected 4-manifolds with prescribed fundamental groups.
method Construction of infinite sets of pairwise smoothly inequivalent 2-spheres that are topologically isotopic.
result First known examples of knotting phenomena in 4-manifolds with specific properties.
The paper introduces a group LSP of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the LSP-groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery ob…
Classifies hyperbolic manifolds with specific automorphism groups.
problem Classifying Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups.
method Analyzes manifolds of dimension n≥2 with automorphism groups of dimensions n2−7 or n2−8. result Completes the classification for automorphism groups n2−7 and n2−8. A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…
Holonomy group of Bismut connection on Vaisman manifolds is studied.
problem Analyzing the holonomy group of Bismut connection on Vaisman manifolds.
method Proved and computed the holonomy group for Vaisman manifolds, including solvmanifolds and Hopf manifolds.
result Holonomy group of Bismut connection on Vaisman manifolds is contained in U(n-1).
New definite 4-manifolds found with non-cyclic groups.
problem Finding exotic smooth structures on 4-manifolds with specific fundamental groups.
method Constructing infinitely many non-diffeomorphic structures.
result Infinitely many pairwise non-diffeomorphic definite 4-manifolds with Z/2imesZ/2 fundamental group.