Finite group actions on smooth 3-manifolds can be smoothed.
problem Finite group actions on smooth 3-manifolds.
method Uniform limit of smooth actions.
result Every continuous action of a finite group on a smooth 3-manifold is a uniform limit of smooth actions.
Surveying matrix group actions on manifolds.
problem Topological Zimmer's conjecture on matrix group actions on manifolds.
method Surveying existing research and literature.
result Status update on matrix group actions on manifolds.
Survey on finite group actions on manifolds.
problem Understanding actions of finite groups on manifolds.
method Analyzes the restriction of actions to subgroups and considers group and manifold properties.
result Provides information on subgroup actions and their properties.
Researchers calculate the action dimension of various group structures.
problem Determining the minimum dimension of contractible manifolds with group actions.
method Computing the action dimension of specific group structures.
result Action dimensions computed for Artin groups, graph products, and aspherical complements.
The study explores large group actions on 3-manifolds and their Heegaard splittings.
problem Understanding finite group actions on 3-manifolds and their properties.
method Investigates maximal group orders and presents specific examples of 3-manifolds with large actions.
result The existence of a unique hyperbolic 3-manifold with a specific large group action.
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.
The study extends group actions from surfaces to 3-manifolds using G-cobordisms.
problem When a finite group action on a surface extends to a 3-manifold.
method Using Schur multiplier and G-cobordisms, the study examines principal actions. result Affirmative extension for abelian, dihedral, symmetric, and alternating groups.
Smooth actions of cyclic groups on 3-manifolds are conjugate to smooth ones.
problem Understanding smoothness in group actions on 3-manifolds.
method Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds. result Finite cyclic group actions by (1+ε)-bilipschitz homeomorphisms on closed 3-manifolds are conjugate to smooth actions. A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to in…
Study explores new actions on product manifolds with asymmetric factors.
problem Exploring effective circle actions on product manifolds with asymmetric factors.
method Proves existence of infinite families of distinct non-diagonal effective circle actions on products of asymmetric manifolds with Sn. result Infinite family of distinct non-diagonal effective circle actions on MimesS2. The study explores finite group actions on elliptic 3-manifolds.
problem Finite, fiber-preserving group actions on elliptic 3-manifolds.
method General construction and restriction to elliptic 3-manifolds, proof for orientation-reversing and fiber-preserving diffeomorphisms, analysis of each type of elliptic 3-manifold.
result A presentation for the quotient space under finite group actions is constructed.
Study actions of homeomorphism groups on manifolds with specific properties.
problem Actions of homeomorphism groups on manifolds with nontrivial finite free actions.
method Analyzes the action of extHomeo0(M) on another manifold N under certain conditions. result If M is not an Fp-homology sphere, then N≅MimesK for a homology manifold K. Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. The paper studies finite group actions on Seifert manifolds, proving conditions for their existence.
problem Finite group actions on Seifert manifolds that preserve fiber and orientation.
method Constructs group actions and refines obstruction conditions.
result Provides general structure of finite groups that can act on Seifert manifolds.
Study shows groups can act on torus without extending to 3-manifold.
problem Bordism problem for group actions on torus.
method Examples of groups acting on torus by diffeomorphisms isotopic to identity.
result Groups cannot be extended to action on bounding 3-manifold.
Groups can act on 3-manifolds if their Cayley complex can embed in specific types of 3-manifolds.
problem Understanding group actions on 3-manifolds.
method Proving groups admit actions on 3-manifolds if their Cayley complexes can embed in specific 3-manifolds.
result Groups with embeddable Cayley complexes can act on four specific types of 3-manifolds.
A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental d…
Example of group action on surface that can't extend to 3-manifold.
problem Finding group actions on surfaces that can't be extended to 3-manifolds.
method Provided a first known example and sufficient conditions for infinitely many such actions.
result Finite group actions on surfaces that are free and orientation-preserving do not extend to arbitrary actions on 3-manifolds.
Study geodesics on orbifolds and manifolds with group actions.
problem Existence of closed geodesics on orbifolds and manifolds.
method Analyzes geodesics on compact and noncompact manifolds with specific group actions.
result Showed existence of nontrivial closed geodesics on certain 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.
The study characterizes group actions on 3-manifolds achieving maximal order.
problem Understanding maximal group actions on 3-manifolds.
method Analyzing finite group-actions on handlebodies and 3-manifolds.
result Characterization of 3-manifolds and groups achieving maximal order.
Study shows cohomological limits on extending actions on 3-manifold boundaries.
problem Limits to extending group actions on 3-manifold boundaries.
method Cohomological obstructions for C0-actions on 3-manifolds. result Cohomological obstructions prevent extending actions on certain 3-manifold boundaries.
Study finite group actions on 4-manifolds, finding rank bounds.
problem Understanding finite group actions on 4-manifolds.
method Investigate Borel spectral sequence for G-equivariant cohomology.
result Establish new bounds on the rank of G for homologically trivial actions.
Develops a spectral sequence for Lie group actions on manifolds.
problem Understanding cohomology of manifolds with Lie group actions.
method Introduces a spectral sequence relating manifold cohomology to Lie algebra cohomology.
result Establishes a new description of de Rham cohomology for manifolds with Lie group actions.
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.
The study finds obstructions to smooth group actions on 4-manifolds using Seiberg-Witten theory.
problem Obstacles to smooth group actions on 4-manifolds.
method Families Seiberg-Witten theory to find obstructions.
result Certain group actions on H2(X,Z) can't be lifted to smooth actions on X. Examples show cyclic actions on 4-manifolds can't extend to Hamiltonian circle actions.
problem Cyclic group actions on 4-manifolds that trivialize homology cannot be extended to Hamiltonian circle actions.
method Holomorphic methods applied to combinatorial tools for circle actions.
result Cyclic actions on 4-manifolds that trivialize homology do not extend to Hamiltonian circle actions.
Reduction principles for proper actions on smooth manifolds.
problem Proper actions on smooth manifolds and their properties.
method Exhibit constructions and prove reduction principles for proper actions.
result Reduction principles hold for proper actions, polar actions, and copolarity.
The study examines group actions of SL_n(Z) on aspherical manifolds and their implications.
problem Understanding group actions of SL_n(Z) on aspherical manifolds.
method Analyzing the induced group homomorphisms and holonomy groups.
result The group SL_n(Z) cannot act nontrivially on aspherical manifolds when the dimension is less than the group's rank.
The paper classifies actions of homeomorphism groups on manifolds.
problem Classifying actions of homeomorphism groups on manifolds.
method General classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds.
result Definitive answers to conjectures and resolutions of many problems in Ghys' program.
Finite actions of lattices on manifolds proven for certain groups.
problem Finite actions of lattices on compact manifolds.
method Uses machinery from Brown, Fisher, and Hurtado.
result Finite actions proven for lattices in p-adic and S-arithmetic groups. Geodesic surfaces embed into hyperbolic 3-manifolds for all finite group actions.
problem Embedding geodesic surfaces into hyperbolic 3-manifolds.
method Analyzing finite group actions on surfaces and proving geodesic embeddings for all irreducible cases.
result All quasiplatonic surfaces embed geodesically into hyperbolic 3-manifolds.
Paper constructs Witten and elliptic genera for noncompact manifolds with proper actions.
problem Constructing genera for noncompact manifolds with proper actions.
method Using proper cocompact actions by almost connected Lie groups, the paper constructs and proves properties of Witten and elliptic genera.
result Vanishing and rigidity results generalizing known compact group actions.
Infinite 2-groups of bounded exponent cannot act faithfully on compact manifolds.
problem Actions of infinite 2-groups of bounded exponent on compact manifolds.
method Analyzing properties of 2-groups of bounded exponent.
result Infinite 2-groups of bounded exponent cannot act faithfully on compact manifolds.
Study unipotent group actions on Kähler manifolds using moment maps.
problem Analyzing unipotent group actions on compact Kähler manifolds.
method Moment map techniques, stability conditions, symplectic reduction.
result Geometric quotients with compactifiable Kähler structures.
Study of pseudo-Riemannian manifolds with M{ö}bius group actions.
problem Characterizing pseudo-Riemannian manifolds with specific group actions.
method Analyzing the action of the conformal group on compact manifolds.
result If the non-compact semi-simple part of the conformal group is M{ö}bius, the manifold is conformally flat.
New findings on symmetries of 4-manifolds and actions of automorphism groups.
problem Characterizing symmetries of simply-connected 4-manifolds and automorphism groups of free groups.
method Analyzing actions of compact Lie groups and automorphism groups of free groups on 4-manifolds.
result Actions of certain groups on 4-manifolds are trivial or factor through a specific group.
Reduces proper actions to simpler core actions for analysis.
problem Understanding properties of proper actions on manifolds.
method Extending Skjelbred and Straume's construction to non-compact groups, focusing on core of actions.
result Properties of proper actions are determined by simpler core actions.
The paper classifies multiplicity free Hamiltonian actions and quasi-Hamiltonian manifolds.
problem Classifying multiplicity free Hamiltonian actions and quasi-Hamiltonian manifolds.
method Classification through symplectic reductions and actions on loop groups.
result Recovery of old and new examples of multiplicity free structures.
Study finite group actions on manifolds with non-zero degree maps to nilmanifolds.
problem Understanding finite group actions on manifolds with specific properties.
method Analyzes effective actions, bounds discrete degrees, studies toral rank conjecture, and introduces iterated actions.
result Proves Homeo(M) is Jordan and bounds the discrete degree of symmetry.
Conditions for equivariant bundles on 4-manifolds with cyclic actions.
problem Existence of equivariant bundles on 4-manifolds with cyclic actions.
method Conditions derived from the twisted signature formula and congruence relations between fixed point data and isotropy representations.
result Necessary and sufficient conditions for the existence of equivariant bundles.
We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.
Study shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
We study discrete, cocompact, isometric actions of groups on Hadamard spaces, and the induced actions on ideal boundaries. For a class of groups generalizing fundamental groups of three-dimensional graph manifolds, we find a set of invariants for the action which determine the boundary action up to equivariant homeomor…
We study group actions on manifolds that admit hierarchies, which generalizes the idea of Haken n-manifolds introduced by Foozwell and Rubinstein. We show that these manifolds satisfy the Singer conjecture in dimensions n≤4. Our main application is to Coxeter groups whose Davis complexes are manifolds; we show th…
Survey on collapsing manifolds using group actions and foliations.
problem Collapsing manifolds with controlled curvature.
method Using group actions and singular Riemannian foliations.
result Recent extensions to singular Riemannian foliations.
New 4-manifolds with distinct smoothings via group actions.
problem Constructing distinct smooth 4-manifolds.
method Effective G-actions on contractible 4-manifolds with distinct boundary-smoothings.
result Distinct smooth 4-manifolds obtained from contractible 4-manifolds with G-actions.
We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analo…