Automorphisms Of Abelian Groups Research Articles

K3 surface entropy and automorphism groups

We derive a characterization of the complex projective K3 surfaces which have automorphisms of positive entropy in terms of their Néron–Severi lattices. Along the way, we classify the projective K3 surfaces of zero entropy with infinite automorphism groups and we determine the projective K3 surfaces of Picard number at least five with almost abelian automorphism groups, which gives an answer to a long standing question of Nikulin.

Trivializing group actions on braided crossed tensor categories and graded braided tensor categories

For an abelian group $A$, we study a close connection between braided $A$-crossed tensor categories with a trivialization of the $A$-action and $A$-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a tensor category $\mathcal{C}$ is given by an element $O(T) \in H^2(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T) = 0$, the set of obstructions forms a torsor over $\operatorname{Hom}(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$, where $\operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed Etingof et al., allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided $A$-crossed tensor category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $\mathbb{Z}/2\mathbb{Z}$-crossed structures over Tambara–Yamagami fusion categories and, consequently, a conceptual interpretation of the results by Siehler about the classification of braidings over Tambara–Yamagami categories.

On profinite groups with automorphisms whose fixed points have countable Engel sinks

An Engel sink of an element g of a group G is a set $${\cal E}(g)$$ such that for every x ∈ G all sufficiently long commutators [⋯[[x, g], g],…, g] belong to $${\cal E}(g)$$ . (Thus, g is an Engel element precisely when we can choose $${\cal E}(g) = \{ 1\} $$ .) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q2 for a prime q such that for each a ∈ A {1} every element of the centralizer CG(a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

On abelian automorphism groups of hypersurfaces

Given integers d ≥ 3 and N ≥ 3, let G be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree d in the complex projective space ℙN−1. Suppose G ⊂ PGL(N, ℂ) can be lifted to a subgroup of GL(N, ℂ). Suppose moreover that there exists an element g in G such that G/❬g❭ has order coprime to d − 1. Then all possible G are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given (d, N). In particular, we show (Proposition 5.1) that the order of an automorphism of a smooth cubic fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers is achieved by a unique (up to isomorphism) cubic fourfold.

KMS states on crossed products by abelian groups

We provide a general description of the KMS states for flows whose fixed point algebra satisfies a certain regularity condition. This is then applied to crossed products by discrete groups, and in particular to certain flows on crossed products by discrete abelian groups where the methods can be combined with spectral analysis for abelian automorphism groups.

HOMOGENEOUS STRUCTURES WITH NONUNIVERSAL AUTOMORPHISM GROUPS

AbstractWe present three examples of countable homogeneous structures (also called Fraïssé limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free.

Picard Groups of Moduli Spaces of Curves with Symmetry

Abstract We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the 1st part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the 2nd part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.

Tetravalent vertex- and edge-transitive graphs over doubled cycles

In order to complete (and generalize) results of Gardiner and Praeger on 4-valent symmetric graphs [3] we apply the method of lifting automorphisms in the context of elementary-abelian covering projections. In particular, the vertex- and edge-transitive graphs whose quotient by a normal p-elementary abelian group of automorphisms, for p an odd prime, is a cycle, are described in terms of cyclic and negacyclic codes. Specifically, the symmetry properties of such graphs are derived from certain properties of the generating polynomials of cyclic and negacyclic codes, that is, from divisors of xn±1∈Zp[x]. As an application, a short and unified description of resolved and unresolved cases of Gardiner and Praeger are given.

Groups in which each subgroup is commensurable with a normal subgroup

A group G is a cn-group if for each subgroup H of G there exists a normal subgroup N of G such that the index |HN:(H∩N)| is finite. The class of cn-groups contains properly the classes of core-finite groups and that of groups in which each subgroup has finite index in a normal subgroup.In the present paper it is shown that a cn-group whose periodic images are locally finite is finite-by-abelian-by-finite. Such groups are then described into some details by considering automorphisms of abelian groups. Finally, it is shown that if G is a locally graded group with the property that the above index is bounded independently of H, then G is finite-by-abelian-by-finite.

Note on Caranti’s method of construction of Miller groups

The non-abelian groups with abelian group of automorphisms are widely studied. Following Earnley, such groups are called Miller groups, since the first example of such a group was given by G.A. Miller in 1913. Many other examples of Miller p-groups have been constructed by several authors. Recently, Caranti (Isr J Math 205: 235–246, 2015) provided module theoretic methods for constructing non-special Miller p-groups from special Miller p-groups. By constructing examples, we show that these methods do not always work. We also provide a sufficient condition on special Miller p-group for which the methods of Caranti work.

Erratum to “A module-theoretic approach to abelian automorphism groups”

Erratum to “A module-theoretic approach to abelian automorphism groups”

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let $$\mathcal S$$ be an abelian group of automorphisms of a probability space $$(X, {\mathcal A}, \mu )$$ with a finite system of generators $$(A_1, \ldots , A_d).$$ Let $$A^{{\underline{\ell }}}$$ denote $$A_1^{\ell _1} \ldots A_d^{\ell _d}$$ , for $${{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).$$ If $$(Z_k)$$ is a random walk on $${\mathbb {Z}}^d$$ , one can study the asymptotic distribution of the sums $$\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}$$ and $$\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f$$ , for a function f on X. In particular, given a random walk on commuting matrices in $$SL(\rho , {\mathbb {Z}})$$ or in $${\mathcal M}^*(\rho , {\mathbb {Z}})$$ acting on the torus $${\mathbb {T}}^\rho $$ , $$\rho \ge 1$$ , what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $${\mathbb {T}}^\rho $$ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), $$\mathcal S$$ a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.

Mixing properties of commuting nilmanifold automorphisms

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.

Drinfeld centers for bicategories

We generalize Drinfeld's notion of the center of a tensor category to bicategories. In this generality, we present a spectral sequence to compute the basic invariants of Drinfeld centers: the abelian monoid of isomorphism classes of objects, and the abelian automorphism group of its identity object. There is an associated obstruction theory that explains the difference between the Drinfeld center and the center of the classifying category. For examples, we discuss bicategories of groups and bands, rings and bimodules, as well as fusion categories.

On the Gonality of an Algebraic Curve and Its Abelian Automorphism Groups

Let G be an abelian automorphism group of a complex algebraic curve of genus g ≥ 2 with gonality d, and let |G| be its order. We prove that except for the Fermat curve of degree d + 1, and if G is cyclic. Furthermore, if |G| > 2g − 2 + 2d and C admits a unique finite cover f: C → ℙ1 of degree d, then f is a Galois cover.

A module-theoretic approach to abelian automorphism groups

There are several examples in the literature of finite non-abelian $p$-groups whose automorphism group is abelian. For some time only examples that were special $p$-groups were known, until Jain and Yadav [JY12] and Jain, Rai and Yadav [JRY13] constructed several non-special examples. In this paper we show how a simple module-theoretic approach allows the construction of non-special examples, starting from special ones constructed by several authors, while at the same time avoiding further direct calculations.

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,ℝ)×…×SL(2,ℝ)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Livšic’s theorem for Anosov flows for a standard partially hyperbolic ℝ d - or ℤ d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,ℝ) d /Γ and are a step in the direction of the “top-degree cohomology” part.

ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP

We construct, for the first time, various types of specific non-special finite p-groups having abelian automorphism group. More specifically, we construct groups G with abelian automorphism group such that γ2(G) < Z(G) < Φ(G), where γ2(G), Z(G) and Φ(G) denote the commutator subgroup, the center and the Frattini subgroup of G respectively. For a finite p-group G with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) Z(G) = Φ(G) is elementary abelian; (ii) γ2(G) = Φ(G) is elementary abelian, where p is an odd prime. We construct examples to show the existence of groups G with elementary abelian automorphism group for which exactly one of the above two conditions holds true.

Model domains in ℂ3 with abelian automorphism group

It is shown that every hyperbolic rigid polynomial domain in ℂ3 of finite-type, with abelian automorphism group is equivalent to a domain that is balanced with respect to some weight.

On the number of rational points on curves over finite fields with many automorphisms

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin–Schreier curves of the form yq−y=f(x) with f∈Fqr[x], on which the additive group Fq acts, and Kummer curves of the form yq−1e=f(x), which have an action of the multiplicative group Fq⋆. In both cases we can remove a q factor from the Weil bound when q is sufficiently large.