Finite Group Of Automorphisms Research Articles

Algebraically hyperbolic manifolds have finite automorphism groups

A projective manifold [Formula: see text] is algebraically hyperbolic if there exists a positive constant [Formula: see text] such that the degree of any curve of genus [Formula: see text] on [Formula: see text] is bounded from above by [Formula: see text]. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Compressible Finite Groups of Birational Automorphisms

The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established.

On the fundamental group of a smooth projective surface with a finite group of automorphisms

In this article we prove new results on fundamental groups for some classes of fibered smooth projective algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of the surfaces under study upto finite index. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori’s well-known question on fundamental groups and free abelianness of second homotopy groups for these surfaces. We also prove a theorem that bounds the multiplicity of the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms $G$ in terms of the multiplicities of the induced fibration on $X/G$. If $X/G$ is a $\mathbb{P}^1$-fibration, we show that the multiplicity actually divides $|G|$. This theorem on multiplicity, which is of independent interest, plays an important role in our theorems.

Almost nilpotency of an associative algebra with an almost nilpotent fixed-point subalgebra

Let A be an associative algebra of arbitrary dimension over a field F and G a finite group of automorphisms of A of order n, prime to the characteristic of F. Denote by AG={a∈A|ag=afor allg∈G} the fixed-point subalgebra. By the classical Bergman–Isaacs theorem, if AG is nilpotent of index d, i.e. (AG)d=0, then A is also nilpotent and its nilpotency index is bounded by a function depending only on n and d. We prove, under the additional assumption of solubility of G, that if AG contains a two-sided nilpotent ideal I◁AG of nilpotency index d and of finite codimension m in AG, then A contains a nilpotent two-sided ideal H◁A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d. An even stronger result is provided for graded associative algebras: if G is a finite (not necessarily soluble) group of order n and A=⨁g∈GAg is a G-graded associative algebra over a field F, i.e. AgAh⊂Agh, such that the identity component Ae has a two-sided nilpotent ideal Ie◁AG of nilpotency index d and of finite codimension m in Ae, then A has a homogeneous nilpotent two-sided ideal H◁A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of n, d and m.

Nielsen Realization by Gluing: Limit Groups and Free Products

We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings’ theorem on groups with at least two ends, in which some control over the behavior of virtual free factors is gained.

Quotients of del Pezzo surfaces of high degree

In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field k \Bbbk of characteristic zero by finite groups of automorphisms. We show that if a del Pezzo surface X X contains a point defined over the ground field and the degree of X X is at least five, then the quotient is always k \Bbbk -rational. If the degree of X X is equal to four, then the quotient can be non- k \Bbbk -rational only if the order of the group is 1 1 , 2 2 , or 4 4 . For these groups we construct examples of non- k \Bbbk -rational quotients.

On autocommutators and the autocommutator subgroup in infinite abelian groups

Abstract It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

On orbifold theory

Let V be a simple vertex operator algebra and G a finite automorphism group of V such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined and a global dimension formula for V in terms of twisted modules is obtained. In particular, the orbifold theory conjecture is completely solved if G is solvable.

Congruence property in orbifold theory

Let V V be a rational, selfdual, C 2 C_2 -cofinite vertex operator algebra of CFT type, and G G a finite automorphism group of V . V. It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to V V and G G is a congruence subgroup. In particular, the q q -character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case V V is the Frenkel-Lepowsky-Meurman’s moonshine vertex operator algebra and G G is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in the monster group is a modular function.

Enriques surfaces in characteristic 2 with a finite group of automorphisms

Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a 1-dimensional family of classical and supersingular Enriques surfaces with the automorphism group A u t ( X ) \mathrm {Aut}(X) isomorphic to the symmetric group S 5 \mathfrak {S}_5 of degree 5.

Gradings on Lie algebras with applications to infra-nilmanifolds

In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. The existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier and we give a different proof of this fact. Similarly, we prove that given a grading of one of these types and a finite group of automorphisms, there always exist a grading of the same type which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphism but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type.

Explicit Galois representations of automorphisms on holomorphic differentials in characteristic p

We determine the representation of the Galois group for the cyclotomic function fields in characteristic p>0 induced by the natural action on the space of holomorphic differentials via construction of an explicit canonical basis of differentials. This includes those cases which present wild ramification and finite automorphism groups with non-cyclic p-part, which have remained elusive. We also obtain information on the gap sequences of ramified primes and extend these results to rank one Drinfel'd modules.

Smooth quotients of bi-elliptic surfaces

We consider the quotient X of bi-elliptic surface by a finite automorphism group. If X is smooth, then it is a bi-elliptic surface or ruled surface with irregularity one. As a corollary any bi-elliptic surface cannot be Galois covering of projective plane, hence does not have any Galois embedding.

A generalization of Drygas functional equation

We obtain the Solutions of the following Drygas functional equation∑ λ ∈Φ f (x + λy + aλ ) = κf(x)+ ∑ λ ∈Φ f(λy), x, y ∈ Swhere S is an abelian semigroup, G is an abelian group, f ∈ GS, Φ is a finite automorphism group of S with order k, and aλ ∈ S, λ∈Φ.

On automorphisms of the punctual Hilbert schemes of K3 surfaces

We present a sufficient condition for the punctural Hilbert scheme of length two of a K3 surface with finite automorphism group to have automorphism group of infinite order in geometric terms (Theorem 2.1) and supply it with concrete examples (Theorem 1.2). We also discuss Mori dream space structures under an extermal crepant resolution (Theorems 1.2, 4.1, 5.2) from the viewpoint of automorphisms. These affirmatively answer a question of Malte Wandel.

Towards a classification of symplectic automorphisms on manifolds of $$K3^{[n]}$$ K 3 [ n ] type

The present paper is devoted to the classification of symplectic automorphisms of some hyperkahlermanifolds. The result presented here is a proof that all finite groups of symplectic automorphisms of manifolds of \(K3^{[n]}\) typeare contained in Conway’s group \(Co_1\).

Combinatorial categories and permutation groups

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ , with automorphism group isomorphic to Γ / N . It is shown how to enumerate such objects with a given finite automorphism group G , how to represent them all as quotients of a single regular object U ( G ) , and how the outer automorphism group of Γ acts on them. Examples constructed include kaleidoscopic maps with trinity symmetry.

QUOTIENTS OF CONIC BUNDLES

Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group $$ {\mathrm{\mathfrak{C}}}_{2^k} $$ of order 2 k , dihedral group $$ {\mathfrak{D}}_{2^k} $$ of order 2 k , alternating group $$ {\mathfrak{A}}_4 $$ of degree 4, symmetric group $$ {\mathfrak{S}}_4 $$ of degree 4 or alternating group $$ {\mathfrak{A}}_5 $$ of degree 5 effectively acting on the base of the conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.

The structure of C*-subalgebras of the Toeplitz algebra fixed with respect to a finite group of automorphisms

In this paper we study the C*-subalgebras of the Toeplitz algebra T, each element of which is fixed relative to finite subgroup of automorphisms of the algebra T. We prove that such subalgebras have a finite family of unitarily equivalent irreducible representations.

The structure of invariant ideals of some subalgebras of Toeplitz algebra

In this paper we give a complete description of invariant ideals of C*-subalgebras of Toeplitz algebra that are fixed with respect to a finite group of automorphisms. The splitting property of short exact sequences generated by such ideals is established.