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.
Finite group action on K-stability results in standard stability.
problem Understanding K-stability under finite group action.
method Analyzing G-equivariant K-semistability and K-polystability for log Fano pairs.
result G-equivariant K-semistability implies K-semistability for log Fano pairs.
The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remai…
New findings on group actions and stabilizers of infinite sets.
problem Understanding finiteness properties of stabilizers in group actions.
method Analyzing oligomorphic actions and permutational wreath products.
result Existence of finite subsets with non-FP∞ stabilizers. We consider orientation-preserving actions of a finite group G on the 3-sphere S^3 (and also on Euclidean space R^3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal …
Convex cores found for group actions on median spaces.
problem Understanding group actions on median spaces without metric or topology.
method Introduced convex cores for actions on finite-rank median algebras.
result Actions on median spaces have nonempty convex cores.
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.
It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each …
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. 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 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.
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.
We show that every continuous action of a finite group on a smooth three-manifold is a uniform limit of smooth actions.
We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R^n-trees. We first prove that Sela's limit groups do have a free action on an R^n-tree. We then prove that a finitely generated group having a free action on an R^n-tree can be obtained from free abelian groups and surfac…
We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of certain orbifold fundamental groups.
A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that…
Vector fields invariant under Lie group action are finitely generated by polynomial fields.
problem Understanding invariant vector fields under Lie group actions.
method Analyzing the module of smooth vector fields invariant under a linear action of a compact Lie group.
result The module of invariant vector fields is finitely generated by polynomial fields.
Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but that…
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…
The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds M. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest grou…
Study finite group actions on Higgs bundle moduli spaces.
problem Describe fixed points of finite group actions on Higgs bundle moduli spaces.
method Use twisted Γ-equivariant bundles and Prym-Narasimhan-Ramanan construction.
result Provide description of fixed-point subvarieties of certain finite group actions on G-character varieties.
Finite index subgroups of certain groups cannot act faithfully on the circle.
problem Finite index subgroups of specific groups cannot act faithfully on the circle.
method Analyzing C1 actions on the circle for finite index subgroups of mapping class groups, automorphism groups, and outer automorphism groups. result No orientation preserving C1 action of finite index subgroups of these groups on the circle can be faithful. Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for …
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
Non-isomorphic groups with similar profinite completions found.
problem Finding non-isomorphic groups with similar profinite completions.
method Exhibited infinitely many pairs of non-isomorphic groups with specific properties.
result Groups with Property FA and non-trivial actions on trees have isomorphic profinite completions.
Finite group action on a surface yields a special homology subspace.
problem Finite group action on a surface.
method Invariant Lagrangian subspace in homology.
result Existence of a G-invariant Lagrangian subspace in H1(S).
Study symplectic and Hamiltonian actions on irrational ruled surfaces, proving existence and non-existence of extensions.
problem Tackles the extension of symplectic and Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic 4-manifolds.
method Constructs symplectic involutions and cyclic actions, classifies symplectic morphisms, and proves non-extendability of certain actions.
result Shows existence and non-existence of Hamiltonian circle actions for different cyclic actions on irrational ruled symplectic 4-manifolds.
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. Any continuous action of SL(n,Z), where n > 2, on a r-dimensional mod 2 homology sphere factors through a finite group action if r < n - 1. In particular, any continuous action of SL(n+2,Z) on the n-dimensional sphere factors through a finite group action.
It is shown that for any action of a finitely presented group G on an R-tree, there is a decomposition of G as the fundamental group of a graph of groups related to this action. If the action of G on T is non-trivial, i.e. there is no global fixed point, then G has a non-trivial action on a simplcial R…
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.
We give bordism-finiteness results for manifolds with semi-simple group action. Consider the class of oriented manifolds which admit a circle action with isolated fixed points such that the action extends to an S3-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler c…
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.
Finite p-group actions on manifolds have limited stabilizer subgroups.
problem Understanding the structure of stabilizer subgroups in group actions on manifolds.
method Bounding the index of a subgroup H in a finite p-group G acting on a compact manifold M, ensuring a controlled number of stabilizers.
result The existence of a subgroup H with a controlled index and limited stabilizers.
This is a survey on upper and lower bounds for finite group actions on bounded surfaces, 3-dimensional handlebodies and closed handles, handlebodies in arbitrary dimensions and finite graphs (the common feature of these objects is that all have free fundamental group).
Finite groups act freely on surfaces but not on 3-manifolds.
problem Understanding finite group actions on surfaces and 3-manifolds.
method Analyzing homeomorphisms of surfaces and 3-manifolds.
result Finite groups can act freely on surfaces but not on handlebodies.
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1),…
Study on finiteness property of right-angled Artin groups actions on extension graphs.
problem Finiteness property of hyperbolic simplicial actions on right-angled Artin groups.
method Analysis of right-angled Artin group actions on extension graphs, using asymptotic translation lengths and syllable lengths.
result Asymptotic translation lengths of elements in right-angled Artin groups are rational and have a common denominator under certain conditions.
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. This paper discusses topological and locally linear actions of finite groups on S4. Local linearity of the orientation preserving actions on S4 forces the group to be a subgroup of SO(5). On the other hand, orientation reversing topological actions of "exotic" groups G (i.e. G⊂O(5)) on S4 are …
New Oliver groups confirm a conjecture about sphere actions.
problem Confirming the Smith question for finite groups acting on spheres.
method Induction of group representations to find new Oliver groups.
result Found an infinite family of Oliver groups satisfying the Laitinen Conjecture.
The paper explores properties of continuous actions on manifolds, proving bounds on subgroup size and fixed points.
problem Properties of continuous finite group actions on topological manifolds.
method Analyzes properties including Jordan property and almost fixed point property, proving bounds on subgroup size.
result Existence of a constant C such that for any continuous action of a finite group G on a manifold X, there is a subgroup H with [G:H] ≤ C and a fixed point.
The paper proves group actions on spheres with odd fixed points.
problem Finite group actions on homology six-spheres with odd Euler characteristics.
method Analyzes smooth actions and fixed point sets of finite groups.
result The group is one of three specific types, and the fixed point set is a single point.
The paper studies groups with proper actions on finite products of hyperbolic spaces.
problem Characterizing groups with proper actions on finite products of hyperbolic spaces.
method Lineal actions and central extensions of groups.
result Groups with property (PH) or (PH') have specific properties related to their actions on hyperbolic spaces.
According to the work of Laitinen, Morimoto, Oliver and Pawałowski, a finite group G has a smooth effective one fixed point action on some sphere if and only if G is an Oliver group. For some finite Oliver groups G of order up to 216, and for G=A5×Cn for n=3,5,7, we present a strategy of excluding o…
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.
We study actions of finitely generated groups on $\bbR$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and a…