Related papers: Counting torsors for wild abelian groups

We study the growth of torsion in the abelianizations of finite index subgroups in finitely generated metabelian groups. This complements earlier work of Kar, Kropholler and the author which covered the finitely presented amenable groups.

We prove that in the function field setting, $\ell$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\ell$-rank growth matches the growth in the setting of genus…

We study the multi-height distribution of rational points of smooth, projective and split toric varieties over $\mathbf{Q}$ using the lift of the number of points to universal torsors.

We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian…

We introduce a new class of Abelian groups which lies strictly between the classes of co-Hopfian groups and Dedekind-finite groups, calling these groups {\it Bassian-finite}. We prove the surprising fact that in the torsion case the…

In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is…

The main purpose of this paper is to describe the abelian part $\mathcal G^{ab}_{K}$ of the absolute Galois group of a global function field $K$ as pro-finite group. We will show that the characteristic $p$ of $K$ and the non $p$-part of…

In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to $\oplus_{p \in P} \Z_p$, where $P$ is the set of prime integers. We…

We prove a Torelli-like theorem for higher-dimensional function fields, from the point of view of "almost-abelian" anabelian geometry.

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…

We obtain upper bounds for the torsion in the $K$-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants.

We present a characterization of cotorsion-free abelian groups in terms of homomorphisms from fundamental groups of Peano continua, which aligns naturally with the generalization of slenderness to non-abelian groups. In the process, we…

We study the non-abelian tensor square $G\otimes G$ for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that $G\otimes G$ is isomorphic to the direct product of…

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is…

We classify maximal tori in groups of type $F_4$ over a local or global field of characteristic different from $2$ and $3$. We prove a local-global principle for embeddings of maximal tori in groups of type $F_4$.

Let R be a Henselian discrete valuation ring with field of fractions K. If X is a smooth variety over K and G a torus over K, then we consider X-torsors under G. If XX/R is a model of X then, using a result of Brahm, we show that X-torsors…

There is a known analogy between growth questions for class groups and for Selmer groups. If $p$ is a prime, then the $p$-torsion of the ideal class group grows unboundedly in $\mathbb{Z}/p\mathbb{Z}$-extensions of a fixed number field $K$,…

We prove that for every ordered abelian group $G$ there exists a non-trivial ordered abelian group $H$ such that $G\preccurlyeq H\oplus G$ with the lexicographic order, and give a first-order characterization of ordered abelian group $G$…

We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\neq p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-group, and $1\to T \to G \to F \to 1$ be an exact sequence of algebraic groups. Extending earlier work of…