Automorphism Group Research Articles

Class number for pseudo-Anosovs

Given two automorphisms of a group G G , one is interested in knowing whether they are conjugate in the automorphism group of G G , or in the abstract commensurator of G G , and how these two properties may differ. When G G is the fundamental group of a closed orientable surface, we present a uniform finiteness theorem for the class of pseudo-Anosov automorphisms. We present an explicit example of a commensurably conjugate pair of pseudo-Anosov automorphisms of a genus 3 3 surface, that are not conjugate in the mapping class group, and we also show that infinitely many pairwise non-commuting pseud-Anosov automorphisms have class number equal to one. In the appendix, we briefly survey the Latimer-MacDuffee theorem that addresses the case of automorphisms of Z n \mathbb {Z}^n , with a point of view that is suited to an analogy with surface group automorphisms.

Liftable automorphisms of right-angled Artin groups

Abstract Given a regular covering map φ : Λ → Γ \varphi\colon\Lambda\to\Gamma of graphs, we investigate the subgroup LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) of the automorphism group Aut ⁡ ( A Γ ) \operatorname{Aut}(A_{\Gamma}) of the right-angled Artin group A Γ A_{\Gamma} . This subgroup comprises all automorphisms that can be lifted to automorphisms of A Λ A_{\Lambda} . We first show that LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) is generated by a finite subset of Laurence’s elementary automorphisms. For the subgroup FAut ⁡ ( φ ) \operatorname{FAut}(\varphi) of Aut ⁡ ( A Λ ) \operatorname{Aut}(A_{\Lambda}) that consists of lifts of automorphisms in LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) , there exists a natural homomorphism FAut ⁡ ( φ ) → LAut ⁡ ( φ ) \operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi) induced by 𝜑. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup IA ⁡ ( A Λ ) \operatorname{IA}(A_{\Lambda}) and deduce a short exact sequence reminiscent of results from the Birman–Hilden theory for surfaces.

The influence of maximal subgroups on Coleman automorphisms

The aim of this paper is to investigate the influence of intrinsic properties of maximal subgroups on Coleman automorphisms of finite groups G. Let M be a maximal subgroup of G. It is proved that if M is simple then Out Col ( G ) = 1 . It is also proved that either Out Col ( G ) is trivial or Out Col ( G ) is an abelian p-group for some prime p provided that M has a unique nontrivial proper normal subgroup. As an application, it is obtained that under some additional conditions the normalizer property holds for such groups.

Isomorphism Problems and Groups of Automorphisms for Ore Extensions K[x][y;fddx] (Prime Characteristic)

Let Λ(f)=K[x][y;fddx] be an Ore extension of a polynomial algebra K[x] over an arbitrary field K of characteristic p>00$$\\end{document}]]> where f∈K[x]. For each polynomial f, the automorphism group of the algebras Λ(f) is explicitly described. The automorphism group AutK(Λ(f))=S⋊Gf is a semidirect product of two explicit groups where Gf is the eigengroup of the polynomial f (the set of all automorphisms of K[x] such that f is their common eigenvector). For each polynomial f, the eigengroup Gf is explicitly described. It is proven that every subgroup of AutK(K[x]) is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra Λ(f) are 2. The prime, completely prime, primitive and maximal ideals of the algebra Λ(f) are classified.

Vector bundle automorphisms preserving Morse-Bott foliations

Let M be a smooth manifold and F a Morse-Bott foliation with a compact critical manifold Σ⊂M. Denote by D(F) the group of diffeomorphisms of M leaving invariant each leaf of F. Under certain assumptions on F it is shown that the computation of the homotopy type of D(F) reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of D(F) consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups D(F) for such foliations are also presented.

Limit Groups and Automorphisms of κ-Existentially Closed Groups

The structure of automorphism groups of κ-existentially closed groups has been studied by Kaya-Kuzucuoğlu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and |Aut(G)|=2κ whenever κ is an inaccessible cardinal and G is the unique κ-existentially closed group of cardinality κ. The cardinality of the automorphism group of a κ-existentially closed group of cardinality λ>κ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case κ=λ namely: If G is a κ-existentially closed group of cardinality κ, then |Aut(G)|=2κ. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal κ, there exist universal groups of cardinality κ.

Isolated torus invariants and automorphism groups of rigid varieties

Isolated torus invariants and automorphism groups of rigid varieties

Modelling three fermion generations with S3 family symmetry within ℂℓ(8)

Abstract We present a model of three fermion generations with SU(3) × U(1) gauge symmetry constructed from the complex Clifford algebra ℂℓ(8), within which the discrete group S 3 acts as a family symmetry. ℂℓ(8) corresponds to the algebra of complex linear maps from the (complexification of the) Cayley-Dickson algebra of sedenions, 𝕊, to itself. The automorphism group of 𝕊 is G 2 × S 3. We interpret S 3, suitably embedded into ℂℓ(8), as a family symmetry. The gauge symmetry SU (3) × U (1) is invariant under S 3. First-generation states are represented in terms of two even ℂℓ(8) semi-spinors, obtained from two minimal left ideals, related to each other via the order-two S 3 symmetry. The remaining two generations are obtained by applying the S 3 symmetry of order-three to the first generation, resulting in three linearly independent generations.

Extended Symmetry of Higher Painlevé Equations of Even Periodicity and Their Rational Solutions

The structure of the extended affine Weyl symmetry group of higher Painlevé equations of N periodicity depends on whether N is even or odd. We find that for even N, the symmetry group A^N−1(1) contains the conventional Bäcklund transformations sj,j=1,…,N, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of N points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to the existence of degenerated solutions, and for N=4, we explicitly show how even reflection automorphisms cause degeneracy of a class of rational solutions obtained on the orbit of the translation operators of A^3(1). We obtain the closed expressions for the solutions and their degenerated counterparts in terms of the determinants of the Kummer polynomials.

Quantum isomorphism of 2-graphs

We formulate a notion of the quantum automorphism group of a [Formula: see text]-graph. After some preliminary computations, we define quantum isomorphism between a pair of [Formula: see text]-graphs. We produce a “nontrivial” example of a pair of [Formula: see text]-graphs that are not quantum isomorphic to each other.

Flag-transitive point-primitive symmetric 2-designs of prime square order

This paper studies flag-transitive, point-primitive automorphism groups of symmetric 2-designs. In particular, if the order of 2-designs is [Formula: see text] with [Formula: see text] prime, we show that the automorphism groups of 2-designs must be of affine or almost simple type.

Hopf Algebra (Co)actions on Rational Functions

In the theory of quantum automorphism groups, one constructs Hopf algebras acting on an algebra K from certain algebra morphisms σ:K→Mn(K). This approach is applied to the field K=k(t) of rational functions, and it is investigated when these actions restrict to actions on the coordinate ring B=k[t2,t3] of the cusp. An explicit example is described in detail and shown to define a new quantum homogeneous space structure on the cusp.

Birational Automorphism Groups of Severi–Brauer Surfaces Over the Field of Rational Numbers

Abstract We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi–Brauer surfaces over the field of rational numbers are $\mathbb{Z}/3\mathbb{Z}$ and $(\mathbb{Z}/3\mathbb{Z})^{2}.$ Moreover, we show that $(\mathbb{Z}/3\mathbb{Z})^{2}$ is contained in $\textrm{Bir}(V)$ for any Severi–Brauer surface $V$ over a field of characteristic different from $2$ and $3$, and $(\mathbb{Z}/3\mathbb{Z})^{3}$ is contained in $\textrm{Bir}(V)$ for any Severi–Brauer surface $V$ over a field of characteristic different from $2$ and $3$, which contains a non-trivial cube root of unity.

Uniform boundedness on rational maps with automorphisms

In this paper, we study the dynamical uniform boundedness conjecture over a family of rational maps with certain nontrivial automorphisms. Specifically, we consider a family of rational maps of an arbitrary degree d≥2 whose automorphism group contains the cyclic group of order d. We prove that a subfamily of this family satisfies the dynamical uniform boundedness conjecture.

Decidability of the isomorphism problem between multidimensional substitutive subshifts

Abstract An important question in dynamical systems is the classification problem, that is, the ability to distinguish between two isomorphic systems. In this work, we study the topological factors between a family of multidimensional substitutive subshifts generated by morphisms with uniform support. We prove that it is decidable to check whether two minimal aperiodic substitutive subshifts are isomorphic. The strategy followed in this work consists of giving a complete description of the factor maps between these subshifts. Then, we deduce some interesting consequences on coalescence, automorphism groups, and the number of aperiodic symbolic factors of substitutive subshifts. We also prove other combinatorial results on these substitutions, such as the decidability of defining a subshift, the computability of the constant of recognizability, and the conjugacy between substitutions with different supports.

Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

Let M be a compact complex manifold, and D⊂M a reduced normal crossing divisor on it, such that the logarithmic tangent bundle TM(−log⁡D) is holomorphically trivial. Let A denote the maximal connected subgroup of the group of all holomorphic automorphisms of M that preserve the divisor D. Take a holomorphic Cartan geometry (EH,Θ) of type (G,H) on M, where H⊂G are complex Lie groups. We prove that (EH,Θ) is isomorphic to (ρ⁎EH,ρ⁎Θ) for every ρ∈A if and only if the principal H–bundle EH admits a logarithmic connection Δ singular on D such that Θ is preserved by the connection Δ.

Automorphism group of a local inclusion graph over the general linear group

The inclusion graph of a group G, written as I n ( G ) , is defined to be an undirected graph with all nontrivial subgroups of G as its vertices, and two distinct subgroups H, K of G are adjacent if either H ≤ K or K ≤ H . A local inclusion graph of G with respect to a given subgroup S, written as I n ( G , S ) , is an induced subgraph of I n ( G ) with vertex set consisting of all nontrivial subgroups of G properly containing S. Indeed, I n ( G ) can be viewed as the special case when S is the identity subgroup. Let GL n ( F ) be the general linear group consisting of all n × n invertible matrices over a field F and T n ( F ) the subgroup of all invertible lower triangular matrices in GL n ( F ) . In this article, an arbitrary automorphism of I n ( GL n ( F ) , T n ( F ) ) is determined and it is proved that the automorphism group of I n ( GL n ( F ) , T n ( F ) ) is isomorphic to the direct product of S n − 1 and S 2 , where S n − 1 is the symmetric group of degree n − 1 .

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

We define quantum automorphism groups of a wide range of discrete structures. The central tool for their construction is a generalisation of the Tannaka-Krein reconstruction theorem. For any direct sum of matrix algebras M, and any concrete unitary 2-category of finite type Hilbert-M-bimodules C, under reasonable conditions, we construct an algebraic quantum group G which acts on M by a action α, such that the category of α-equivariant representations of G on finite type Hilbert-M-bimodules is equivalent to C. Moreover, we explicitly describe how to get such categories from connected locally finite discrete structures. As an example, we define the quantum automorphism group of a quantum Cayley graph.

On the homotopy groups of the automorphism groups of Cuntz–Krieger algebras

In this paper, we first present the homotopy groups of the automorphism groups of Cuntz–Krieger algebras in terms of the underlying matrices defining the Cuntz–Krieger algebras. We also show that the homotopy groups are complete invariants of the isomorphism classes of Cuntz–Krieger algebras. As a result, the isomorphism class of a Cuntz–Krieger algebra is completely determined by the group structure of its weak extension group and strong extension group.

Automorphism groups of bordered non-orientable Klein surfaces of topological genus 4

Automorphism groups of bordered non-orientable Klein surfaces of topological genus 4