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For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…

Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal…

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$.…

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm…

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that…

In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we…

In 2012, Zilber used model-theoretic techniques to show that a curve of high genus over an algebraically closed field is determined by its Jacobian (viewed only as an abstract group with a distinguished subset for an image of the curve). In…

We prove that the torsion subgroup of the abelian fundamental group is finite for a regular geometrically integral projective variety over a local field. We also study the structure of $SK_1(X)$ for a regular projective variety $X$ over a…

We study the set of Dedekind cuts over a linearly ordered Abelian group as a structure over the language (0,<,+,-). Moreover, we obtain a simple set of axioms for the universal part of the theory of such structures. Finally, we prove that…

In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…

An abelian cover is a finite morphism $X\to Y$ of varieties which is the quotient map for a generically faithful action of a finite abelian group $G$. Abelian covers with $Y$ smooth and $X$ normal were studied in…

We extend the usual projective Abel-Radon transform to the larger context of a smooth complete toric variety X. We define and study toric concavity attached to an algebraic splitting vector bundle on X and we prove a toric version of the…

We prove several results on the model theory of Artin groups, focusing on Artin groups which are ``far from right-angled Artin groups''. The first result is that if $\mathcal{C}$ is a class of Artin groups whose irreducible components are…

Let $\cac$ be a smooth projective curve defined over a number field $k$, $A/k(\cac)$ an abelian variety and $(\tau,B)$ the $k(\cac)/k$-trace of $A$. We estimate how the rank of $A(k(\cac))/\tau B(k)$ varies when we take a finite cover…

Let $A, B$ be finite subsets of a torsion-free group $G$. We prove that for every positive integer $k$ there is a $c(k)$ such that if $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in…

Some results that are true in classical groups are investigated in generalized groups and are shown to be either generally true in generalized groups or true in some special types of generalized groups. Also, it is shown that a Bol groupoid…

The existence of invariant transversals for a normal subgroup $H$ in a group $G$ is investigated. This yields counterexamples to a conjecture in case $H$ is abelian and $G$ is finite.

We present some Zermelo-Fraenkel consistency results regarding bi-orderability of groups, as well as a construction of groups with Conradian orders whose every action on metric spaces has bounded orbits. A classical consequence of the…

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine…

Let $K$ be a number field, and let $A$ be an Abelian variety over $K$ which has no CM isogeny-factors over $\overline{K}$. We prove that $A$ has only finitely many torsion points over the maximal $n$-step-solvable extension of $K$ for any…