Group Morphism Research Articles

Etale descent obstruction and anabelian geometry of curves over finite fields

Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the \'etale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve $C_K$ over the global function field $K = F(D)$. Specifically, we show that there is a bijection between the set of conjugacy classes of well-behaved morphism of fundamental groups and locally constant adelic points of $C_K$ that survive \'etale descent. We use this to provide further evidence for the anabelian conjecture by relating it to another recent conjecture by Sutherland and the second author.

Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

An extended derivation (endomorphism) of a (restricted) Lie algebra L L is an assignment of a derivation (respectively) of L ′ L’ for any (restricted) Lie morphism f : L → L ′ f:L\to L’ , functorial in f f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L ′ L’ to every f f ; and (b) if L L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms.

Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups

We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups f:(Ξ,Ξ∞)→(Λ,Λ∞), stating that asdimΞ≤asdimΛ+asdim([ker⁡f]c). This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group G, asdim G is equal to the supremum of asymptotic dimensions of finitely generated subgroups of G. Our version states that, if (Λ,Λ∞) is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of Λ∞, with these approximate subgroups contained in Λ2.

Quotients of skew morphisms of cyclic groups

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

Inducing braces and Hopf Galois structures

Let p be a prime number and let n be an integer not divisible by p and such that every group of order np has a normal subgroup of order p. (This holds in particular for p>n.) Under these hypotheses, we obtain a one-to-one correspondence between the isomorphism classes of braces of size np and the set of pairs (Bn,[τ]), where Bn runs over the isomorphism classes of braces of size n and [τ] runs over the classes of group morphisms from the multiplicative group of Bn to Zp⁎ under a certain equivalence relation. This correspondence gives the classification of braces of size np from the one of braces of size n. From this result we derive a formula giving the number of Hopf Galois structures of abelian type Zp×E on a Galois extension of degree np in terms of the number of Hopf Galois structures of abelian type E on a Galois extension of degree n. For a prime number p≥7, we apply the obtained results to describe all left braces of size 12p and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree 12p.

Skew product groups for monolithic groups

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.

Strongly scale-invariant virtually polycyclic groups

A finitely generated group \Gamma is called strongly scale-invariant if there exists an injective endomorphism \varphi: \Gamma \to \Gamma with the image \varphi(\Gamma) of finite index in \Gamma and the subgroup \bigcap_{n>0}\varphi^n(\Gamma) finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this paper, we study this question for the class of virtually polycyclic groups, i.e. the virtually solvable groups for which every subgroup is finitely generated. Using the \mathbb{Q} -algebraic hull, which allows us to extend the injective endomorphisms of certain virtually polycyclic groups to a linear algebraic group, we show that the existence of such an endomorphism implies that the group is virtually nilpotent. Moreover, we fully characterize which virtually nilpotent groups have a morphism satisfying the condition above, related to the existence of a positive grading on the corresponding radicable nilpotent group. As another application of the methods, we generalize a result of Fel’shtyn and Lee about which maps on infra-solvmanifolds can have finite Reidemeister number for all iterates.

Smooth skew morphisms of dicyclic groups

Smooth skew morphisms of dicyclic groups

Unambiguous injective morphisms in free groups

A morphism g is ambiguous with respect to a word u if there exists a second morphism h≠g such that g(u)=h(u). Otherwise g is unambiguous with respect to u. Thus unambiguous morphisms are those for which the structure of the morphism is preserved in the image. Ambiguity has so far been studied for morphisms of free monoids, where several characterisations exist for the set of words u permitting an (injective) unambiguous morphism. In the present paper, we consider ambiguity of morphisms of free groups, and consider possible analogies to the existing characterisations in the free monoid. While a direct generalisation results in a trivial situation where all morphisms are ambiguous, we discuss some natural and well-motivated reformulations, and provide a characterisation of words in a free group that permit a morphism which is “as unambiguous as possible”.

Generalization of Fourier Transform and Weyl Calculus

In this paper, a surjective morphism of the topological groups from the real line R to the p -curve Cp is introduced, this function maps from the real line to the p -curve on the complex and when p = 2 then coincide with a classical exponent. The properties of p -Fourier transform is studied. The generalization of the Weyl functional calculus is considered.

Braided Galois objects and Sweedler cohomology of certain Radford biproducts

We construct the group of H-Galois objects for a flat and cocommutative Hopf algebra in a braided monoidal category with equalizers provided that a certain assumption on the braiding is fulfilled. We show that it is a subgroup of the group of BiGalois objects of Schauenburg, and prove that the latter group is isomorphic to the semidirect product of the group of Hopf automorphisms of H and the group of H-Galois objects. Dropping the assumption on the braiding, we construct the group of H-Galois objects with normal basis. For H cocommutative we construct Sweedler cohomology and prove that the second cohomology group is isomorphic to the group of H-Galois objects with normal basis. We construct the Picard group of invertible H-comodules for a flat and cocommutative Hopf algebra H. We show that every H-Galois object is an invertible H-comodule, yielding a group morphism from the group of H-Galois objects to the Picard group of H. A short exact sequence is constructed relating the second cohomology group and the two latter groups, under the above mentioned assumption on the braiding. We show how our constructions generalize some results for modules over commutative rings, and some other known for symmetric monoidal categories. Examples of Hopf algebras are discussed for which we compute the second cohomology group and the group of Galois objects.

NIL-AFFINE CRYSTALLOGRAPHIC ACTIONS OF VIRTUALLY POLYCYCLIC GROUPS

A classical result by K. B. Lee states that every group morphism between almost crystallographic groups is induced by an affine map on the nilpotent Lie group whereon these groups by definition act. It is the main technique for studying morphisms between virtually nilpotent groups, having important applications in fixed point theory, for example as a tool to compute Nielsen numbers for self-maps on infra-nilmanifolds. In this paper, we generalize this result to morphisms between virtually polycyclic groups, which are known to act crystallographically by affine transformations on a nilpotent Lie group. The main method is to study the special class of translation-like actions, which behave well under taking subgroups and restricting the action to invariant right cosets.

Classification of skew morphisms of cyclic groups which are square roots of automorphisms

The auto-index of a skew morphism φ of a finite group A is the smallest positive integer h such that φ h is an automorphism of A . In this paper we develop a theory of auto-index of skew morphisms, and as an application we present a complete classification of skew morphisms of finite cyclic groups which are square roots of automorphisms.

Categorical structures of soft groups

In the current paper, the category of soft groups and soft group homomorphisms is constructed and it is proved that this structure satisfies the category conditions. Also, algebraic properties of some types of soft group morphisms are obtained. Finally, an application is presented as ‘cube’ of soft groups and soft group homomorphisms.

Reflexible complete regular dessins and antibalanced skew morphisms of cyclic groups

A skew morphism of a finite group A is a bijection φ on A fixing the identity element of A and for which there exists an integer-valued function π on A such that φ(ab) = φ(a)φπ(a)(b), for all a, b ∈ A. In addition, if φ − 1(a) = φ(a − 1) − 1, for all a ∈ A, then φ is called antibalanced. In this paper we develop a general theory of antibalanced skew morphisms and establish a one-to-one correspondence between reciprocal pairs of antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of reflexible regular dessins with complete bipartite underlying graphs. As an application, reflexible complete regular dessins are classified.

Residual dimension of nilpotent groups

Abstract The function F G ⁢ ( n ) {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for F N ⁢ ( n ) {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of F N ⁢ ( n ) {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for F N ⁢ ( n ) {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of F N ⁢ ( n ) {\mathrm{F}_{N}(n)} can be fully characterized.

Extended homotopy quantum field theories and their orbifoldization

Based on a detailed definition of extended homotopy quantum field theories we develop a field-theoretic orbifold construction for these theories when the target space is the classifying space of a finite group G, i.e. for G-equivariant topological field theories. More precisely, we use a recently developed bicategorical version of the parallel section functor to associate to an extended equivariant topological field theory an ordinary extended topological field theory. One main motivation is the 3-2-1-dimensional case where our orbifold construction allows us to describe the orbifoldization of equivariant modular categories by a geometric construction. As an important ingredient of this result, we prove that a 3-2-1-dimensional G-equivariant topological field theory yields a G-multimodular category by evaluation on the circle. The orbifold construction is a special case of a pushforward operation along an arbitrary morphism of finite groups and provides a valuable tool for the construction of extended homotopy quantum field theories.

Smooth skew morphisms of the dihedral groups

A skew-morphism $\varphi$ of a finite group $A$ is a permutation on $A$ such that $\varphi(1)=1$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$ where $\pi:A\to\mathbb{Z}_{|\varphi|}$ is an integer function. A skew-morphism is smooth if $\pi(\varphi(x))=\pi(x)$ for all $x\in A$. The concept of smooth skew-morphisms is a generalization of that of $t$-balanced skew-morphisms. The aim of the paper is to develop a general theory of smooth skew-morphisms. As an application we classify smooth skew-morphisms of the dihedral groups.

About a Theorem of Hyman Bass and some other Topics

Let V be a vector space of finite dimension over a field K, and $$V^*$$ the dual space; there is a canonical quadratic form on $$V\oplus V^*$$ , and there is a well known group morphism from $$\mathrm {GL}(V)$$ into $$\mathrm {O}(V\oplus V^{*})$$ . This morphism can be factored through the Clifford-Lipschitz group $$\mathrm {GLip}(V \oplus V^{*})$$ that lies over $$\mathrm {O}(V\oplus V^*)$$ ; but when K is the field $$\mathbb {R}$$ or $$\mathbb {C}$$ , it is not possible to factor it through the smaller group $$\mathrm {Spin}(V\oplus V^*)$$ ; this follows from a theorem published by H. Bass in 1974. It is worth recalling this theorem, and comparing it with an article published by Doran, Hestenes, Sommen and Van Acker in 1993, which asserts that every Lie group can be embedded in a spinorial group. The falseness of this assertion is explained at the end of this article.

Change of grading, injective dimension and dualizing complexes

ABSTRACTLet G,H be groups, φ:G→H a group morphism, and A a G-graded algebra. The morphism φ induces an H-grading on A, and on any G-graded A-module, which thus becomes an H-graded A-module. Given an injective G-graded A-module, we give bounds for its injective dimension when seen as H-graded A-module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.