Action Of Finite Group Research Articles

The real section conjecture and Smith's fixed-point theorem for pro-spaces

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order $p$ whose $p$-torsion cohomology can be killed by finite covers. As an application we derive the section conjecture for the real points of a large class of varieties defined over the field of real numbers and the natural analogue of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers.

Finite Group Actions, Rational fixed Points and Weak Neron Models

Finite Group Actions, Rational fixed Points and Weak Neron Models

Some small orbifolds over polytopes

We introduce some compact orbifolds on which there is a certain finite group action having a simple convex polytope as the orbit space. We compute the orbifold fundamental group and homology groups of these orbifolds. We compute the cohomology rings of these orbifolds when the dimension of the orbifold is even. These orbifolds are intimately related to the notion of small cover.

A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence

We show how the Landau–Ginzburg/Calabi–Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund–Hubsch mirror duality construction to provide an analogue picture featuring all Calabi–Yau hypersurfaces and certain quotients by finite abelian group actions.

Classification of free actions on complete intersections of four quadrics

In this paper we classify all free actions of finite groups on Calabi-Yau complete intersection of 4 quadrics in $\PP^7$, up to projective equivalence. We get some examples of smooth Calabi-Yau threefolds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces.

The local lifting problem for actions of finite groups on curves

Let k be an algebraically closed field of characteristic p>0. We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k[[t]]. To each such φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k. We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic 0 in such a way that X/H and Y/H have the same genus for all subgroups H⊂G. We determine for which G the KGB obstruction of every φ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting φ due to Bertin vanishes for some φ, or for all sufficiently ramified φ. These results provide evidence for the strengthening of Oort’s lifting conjecture which is discussed in [8, Conj. 1.2].

On quasitoric orbifolds

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen-Ruan cohomology groups of an almost complex quasitoric orbifold.

On minimal actions of finite simple groups on homology spheres and Euclidean spaces

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 the linear fractional groups PSL(2, p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.

Classification of K3-surfaces with involution and maximal symplectic symmetry

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces, and a complete classification of K3-surfaces with maximal symplectic symmetry is obtained.

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2 ≤ p ≤ 6 . We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p ≥ 2 .

Some linear actions of finite groups with q′-orbits

Let G be a finite group acting faithfully on a finite vector space M in such a way that the centralizer of every element of M contains a Sylow q -subgroup of G as a central subgroup (for a fixed prime divisor q of | G | with ( q , | M |) = 1). Then G is isomorphic to a subgroup of the semi-linear group on M .

A Lefschetz fixed-point formula for certain orbifold $\mathrm C^*$-algebras

Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of crossed product \mathrm C^* -algebras C_0(X) ⋊ G coming from covariant pairs. Here G is assumed countable, X a manifold, and X ⋊ G cocompact and proper. The formula in question describes the graded trace of the map induced by the automorphism on K-theory of C_0(X) ⋊ G , i.e. the Lefschetz number, in terms of fixed orbits of the spatial map. Each fixed orbit contributes to the Lefschetz number by a formula involving twisted conjugacy classes of the corresponding isotropy group, and a secondary construction that associates, by way of index theory, a group character to any finite group action on a Euclidean space commuting with a given invertible matrix.

Free actions of finite groups on $S^n \times S^n$

Let p be an odd prime. We construct a non-abelian extension Γ of S by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1 × S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p−1 × S2p−1. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.

Group Actions on Quasi-Baer Rings

AbstractA ring R is called quasi-Baer if the right annihilator of every right ideal of R is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to C*-algebras. Various examples to illustrate and delimit our results are provided.

Formal completions of Néron models for algebraic tori

We calculate the formal group law which represents the completion of the N\'eron model of an algebraic torus over the rationals that splits in a tamely ramified abelian extension. As a tools in the proof, we define and give criterions to compute the Weil restriction of a formal group law and the analog of the fixed part of a formal group law with respect to the action of a (finite) group.

The genus level of a group

We introduce the notion of the genus level of a group as a tool to help classify finite conformal group actions on compact Riemann surfaces. We classify all groups of genus level 1 and use our results to outline an algorithm to classify actions of p-groups on compact Riemann surfaces. To illustrate our results, we provide a number of detailed examples.

Dihedral manifold approximate fibrations over the circle

Consider the cyclic group C 2 of order two acting by complex-conjugation on the unit circle S 1. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D ∞ if and only if W is the infinite cyclic cover of a free C 2-manifold M such that M admits a C 2-equivariant manifold approximate fibration to S 1. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for orthogonal actions of finite groups on Euclidean space.

Positivity in power series rings

Abstract We extend and generalize results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not satisfy the transversality condition. Such situations arise naturally when one considers semialgebraic sets invariant under finite group actions.

Conjugacy classes in affine Kac–Moody groups and principal G-bundles over elliptic curves

For a simple complex Lie group G the connected components of the moduli space of semistable G-bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action. In this note we will provide a new proof of this result using the invariant theory of affine Kac–Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G.