Action Of Finite Group Research Articles

On Knots with Cyclic Symmetries

This review paper investigates the relationship between symmetries of knots and links in the three-dimensional sphere, with a focus on cyclic symmetries, and their associated polynomial invariants. It examines the behavior of these polynomials in the case where the knot is set-wise fixed by the action of finite cyclic group on the 3-sphere. The study highlights how these invariants can obstruct the existence of such symmetries, providing valuable insights into the topological properties of these knots. The paper reviews established results on polynomial criteria for periodicity and their extensions to freely periodic links. It also explores generalizations to study the symmetries of virtual knots and spatial graphs.

Equivariant birational geometry of linear actions

In this paper, we study linear actions of finite groups in small dimensions, up to equivariant birationality.

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = ( M , ω ) M=(M,\omega ) be either S 2 × S 2 S^2 \times S^2 or C P 2 # C P 2 ¯ \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form ω \omega . Suppose a finite cyclic group Z n \mathbb {Z}_n is acting effectively on ( M , ω ) (M,\omega ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m ( M , ω ) \mathbb {Z}_n\hookrightarrow Ham(M,\omega ) . In this paper, we investigate the homotopy type of the group S y m p Z n ( M , ω ) Symp^{\mathbb {Z}_n}(M,\omega ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n \mathbb {Z}_n actions satisfying certain inequalities involving the order n n and the symplectic cohomology class [ ω ] [\omega ] , the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n \mathbb {Z}_n -actions on Hirzebruch surfaces.

Finite group actions on abelian groups of finite Morley rank

Finite group actions on abelian groups of finite Morley rank

The binary actions of simple groups with a single conjugacy class of involutions

Abstract We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph Γ ⁢ ( C ) \Gamma(\mathcal{C}) based on the conjugacy class 𝒞 of the group 𝐺, which was introduced in our previous work, and the notion of a strongly embedded subgroup of 𝐺. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.

G-solid rational surfaces

We classify G-solid rational surfaces over the field of complex numbers for finite group actions.

Higher Chow cycles on a family of Kummer surfaces

Abstract We construct a collection of families of higher Chow cycles of type $(2,1)$ on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank $\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.

Residue maps, Azumaya algebras, and buildings

The goal of this note is to give an explicit formula for the residues of twists of the matrix algebra in terms of the twisting cocycle. Combined with the Fixed Point Theorem for actions of finite groups on affine buildings, this leads to a quick proof of the well-known characterization of unramified algebras in terms of Azumaya algebras.

Actions of large finite groups on manifolds

In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group [Formula: see text] on a manifold [Formula: see text], these results provide information on the restriction of the action to a subgroup of [Formula: see text] of index bounded above by a number depending only on [Formula: see text]. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology.

Skew group categories, algebras associated to Cartan matrices and folding of root lattices

For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$ -group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.

On Cyclic Actions of Finite Groups

On Cyclic Actions of Finite Groups

Properly outer and strictly outer actions of finite groups on prime C*-algebras

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper, I define the notion of strictly outer action (similar to the definition for von Neumann factors in [S. Vaes, The unitary implementation of a locally compact group action, J. Funct. Anal. 180 (2001) 426–480]) and prove that for finite groups and prime C*-algebras, it is equivalent to the proper outerness of the action. For finite abelian groups this is equivalent to other relevant properties of the action.

TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES IV: DISCRETENESS AND SECTIONS

Abstract Let $\Sigma $ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro- $\Sigma $ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of $\Sigma $ are invertible. The present paper focuses on the topic of comparison between the theory developed in earlier papers concerning pro- $\Sigma $ fundamental groups and various discrete versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the section conjecture and Grothendieck conjecture. This portion of the theory is purely combinatorial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of (discrete) subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under consideration. These results make possible a fairly straightforward translation, into discrete versions, of pro- $\Sigma $ results obtained in previous papers by the authors. Finally, we discuss a construction that was considered previously by M. Boggi in the discrete case from the point of view of the present paper.

On the homotopy fixed points of Maurer–Cartan spaces with finite group actions

On the homotopy fixed points of Maurer–Cartan spaces with finite group actions

Computing equivalence classes of finite group actions on orientable surfaces

This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface Sg of genus g≥2. With each such action of a group G on Sg one can associate the fundamental group Γ=π(O) of the quotient orbifold O=Sg/G, isomorphic to a Fuchsian group determined completely by orbifold's signature. The Riemann existence theorem reduces the problem of the existence of an action of G on Sg to a purely group-theoretical problem of deciding whether there is an smooth epimorphism mapping the Fuchsian group Γ onto the group G. Using computer algebra systems such as Magma or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus g≥2 becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to topological equivalence. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set of finite group actions on Sg with the corresponding signature. In this paper we derive several results on the topological equivalence of finite group actions on Riemann surfaces. As an application, we derive complete lists of finite group actions of genus g≤8 distinguished up to the topological equivalence.

Artin–Schreier varieties with actions of finite orthogonal groups and Weil representations

We study Artin–Schreier varieties with natural actions of finite orthogonal groups. We compute the characters of the cohomology groups of these varieties as representations of the groups. Our main ingredient is Shinoda’s character formulae on Weil representations for symplectic groups. We give trace formulae for projective hypersurfaces with actions of finite orthogonal groups generalizing Fermat hypersurfaces.

Trace formulae for actions of finite unitary groups on cohomology of Artin–Schreier varieties

Associated to a certain additive polynomial, we introduce an Artin–Schreier variety admitting an action of a finite unitary group. We calculate the character of the cohomology as a representation of a finite unitary group. One of our main ingredients is explicit character formulae for Weil representations of unitary groups due to Gérardin. We give another trace formula for a projective hypersurface admitting an action of a finite unitary group.

On parahoric Hitchin systems over curves

In this paper, we talk about parahoric Hitchin systems over smooth projective curves with the structure group a semisimple simply connected group. We prove the equivalence of parahoric Hitchin systems over the curve with Hitchin systems over a corresponding root stack with a finite cyclic group action determined (up to conjugation) by the parahoric data. And we also show the compatibility of the equivalence with Hitchin maps. We work over an algebraically closed field with a mild assumption of the characteristic.

On α-type (equivariant) cohomology of Hom-pre-Lie algebras

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop equivariant cohomology theory for a Hom-pre-Lie algebra equipped with a finite group action by formulating a proper notion of coefficients system for the equivariant cohomology. We also study the associated formal deformation theory for Hom-pre-Lie algebras in the equivariant context.

Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat–Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive k-group G, every finite subgroup of G(k[t]) is conjugate to a subgroup of G(k). This, in particular, implies that if k is a finite extension of the p-adic field Qp, then the group G(k[t]) has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a short proof of the theorem of Raghunathan–Ramanathan [26] about G-torsors over the affine line.