Action Of Finite Group Research Articles

Derived category invariants andL-series

We relate invariants in derived categories associated to tame actions of finite groups on projective varieties over a finite field to zeros of L-functions

NONVANISHING ELEMENTS FOR BRAUER CHARACTERS

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

Finite group actions on Kervaire manifolds

Let MK4k+2 be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1×S2k+1 of spheres. We show that a finite group of odd order acts freely on MK4k+2 if and only if it acts freely on S2k+1×S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k≠2j−1, then MK4k+2 does not admit any free TOP involutions. Free “exotic” (PL) involutions are constructed on MK30, MK62, and MK126. Each smooth structure on MK30 admits a free Z/2×Z/2 action.

Crossed products by actions of finite groups with the Rokhlin property

We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.

Higher order generalized Euler characteristics and generating series

For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line. We compute the generating series of generalized Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them.

Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

We introduce the Rohlin property and approximate representability for finite group actions on stably projectionless C ∗ ^* -algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra W \mathcal {W} and show some classification results of these actions. This study is based on the work of Izumi, Robert’s classification theorem and Kirchberg’s central sequence C ∗ ^* -algebras.

Finite cyclic group actions with the tracial Rokhlin property

We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of Z/2Z on the 2^{\infty} UHF algebra, whose crossed product is not AF because it has nontrivial K_1-group, has the tracial Rokhlin property, and we give an example of an action of Z/2Z on a simple unital AF algebra which has the tracial Rokhlin property and such that the K_0-group of the crossed product has torsion. In particular, the crossed product of a simple unital AF algebra by an action of Z/2Z with the tracial Rokhlin property need not be AF. We give examples of a tracially approximately representable action of Z/2Z on a simple unital AF algebra which is nontrivial on K_0, and of a tracially approximately representable action of Z/2Z on a simple unital AT algebra with real rank zero which is nontrivial on K_1.

Lifting harmonic morphisms II: Tropical curves and metrized complexes

In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.

Extensions of finite cyclic group actions on bordered surfaces

We study the question of the extendability of the action of a finite cyclic group on a compact bordered Klein surface (either orientable or non-orientable). This extends previous work by the authors for group actions on unbordered surfaces. It is shown that if such a cyclic action is realised by means of a non-maximal NEC signature, then the action always extends. For a given integer g ≥ 2 , we determine the order of the largest cyclic group that acts as the full automorphism group of a bordered surface of algebraic genus g, and the topological type of the surfaces on which the largest action takes place. In addition, we calculate the smallest algebraic genus of a bordered surface on which a given cyclic group acts as the full automorphism group of the surface. For this, we deal separately with orientable and non-orientable surfaces, and we also determine the topological type of the surfaces attaining the bounds.

Rokhlin Dimension and C*-Dynamics

We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of $$\mathcal{Z}$$ -stable C*-algebras, where $$\mathcal{Z}$$ denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve $${\mathcal{Z}}$$ -stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic $${\mathbb{Z}^{d}}$$ -actions on compact metrizable and finite dimensional spaces.

Cyclic covers of affine [formula omitted]-varieties

We consider normal affine T-varieties X endowed with an action of finite abelian group G commuting with the action of T. For such varieties we establish the existence of G-equivariant geometrico-combinatorial presentations in the sense of Altmann and Hausen. As an application, we determine explicit presentations of the Koras–Russell threefolds as bi-cyclic covers of A3 equipped with a hyperbolic Gm-action.

Finite symmetries of S 4

Abstract This paper discusses topological and locally linear actions of finite groups on S 4. Local linearity of the orientation preserving actions on S 4 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 S 4 are constructed, and local linearity and stable smoothability of the actions are studied.

Rokhlin dimension: obstructions and permanence properties

This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on C^* -algebras, introduced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This definition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and \mathcal Z -absorption. We establish K -theoretic obstructions to the existence of actions of finite groups with finite Rokhlin dimension (in the commuting tower version). In particular, we show that there are no actions of any nontrivial finite group on the Jiang-Su algebra or on the Cuntz algebra \mathcal O_{\infty} with finite Rokhlin dimension in this sense.

Monadicity Theorem and Weighted Projective Lines of Tubular Type

We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the equivariantization is compatible with the construction of quotient abelian categories by Serre subcategories. We prove that the equivariantization of the graded module category over a graded ring is equivalent to the graded module category over the same ring but with a different grading. We deduce from these results two equivalences between the category of (equivariant) coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve, where the acting groups are cyclic and the two equivalences are somehow adjoint to each other.

INVARIANT THEORY OF FINITE GROUP ACTIONS ON DOWN-UP ALGEBRAS

We study Artin-Schelter Gorenstein fixed subrings of some Artin-Schelter regular algebras of dimension 2 and 3 under a finite group action and prove a noncommutative version of the Kac-Watanabe and Gordeev theorem for these algebras.

Critical points of invariant functions on closed orientable surfaces

The relationship between critical points of equivariant functions and topological invariants of an equivariant action on closed manifold is an interesting problem. In this paper, we study this relationship for orientation-preserving actions of finite groups $$G$$ on a closed orientable surfaces. We give an elementary, but detailed, description of the behaviour of the gradient field of an equivariant $$C^1$$ -function, we present an elementary, differential, proof of the Riemann–Hurwitz formula and we construct invariant $$C^1$$ -functions with the minimal number of critical orbits. These lead us to show that, with a few exceptions, the equivariant Lusternik–Schnirelmann category of a closed orientable topological surface equals the number of singular orbits of the action.

Every finite group action on a compact 3-manifold preserves infinitely many hyperbolic spatial graphs

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

On finite groups acting on a connected sum of 3-manifolds S2×S1

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 there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on H_g which is quadratic in g. For the proofs we develop a calculus for finite group-actions on H_g, by codifying such actions by handle-orbifolds and finite graphs of finite groups.

Central exact sequences of tensor categories, equivariantization and applications

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we obtain a simple characterization of equivariantizations under actions of finite abelian groups. As an application, we show that if ${\mathcal C}$ is a fusion category and $F: {\mathcal C} \to {\mathcal D}$ is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an equivariantization if $p=2$, or if ${\mathcal C}$ is weakly integral and $p$ is the smallest prime factor of FPdim ${\mathcal C}$.

Differential orbifold K-theory

We construct differential K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in differential orbifold K-theory. Finally, we construct a non-degenerate intersection pairing with values in \mathbb{C}/\mathbb{Z} for the subclass of smooth orbifolds which can be written as global quotients by a finite group action. We construct a real subfunctor of our theory, where the pairing restricts to a non-degenerate \mathbb{R}/\mathbb{Z} -valued pairing.