Related papers: On simple groups definable in some valued fields

It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian…

This paper finds a classification, up-to an isomorphism, of abelian torsion groups realizable as Brauer groups of major types of Henselian valued primarily quasilocal fields with totally indivisible value groups. When $E$ is a quasilocal…

Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or…

We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finite group $G$ then the solvable radical $R$ of…

We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it…

We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give…

We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its…

In this paper we consider gradings by a finite abelian group $G$ on the Lie algebra $\mathfrak{sl}_n(F)$ over an algebraically closed field $F$ of characteristic different from 2 and not dividing $n$.

Given a henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any henselian valuation whose value group is not closed in its divisible hull is definable in the…

This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvable Chermak-Delgado…

We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the definability of the derived subgroup in case that the group is…

We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its…

We firstly show that due to their resplendency ordered henselian valued fields admit relative field quantifier elimination in the Denef--Pas language expanded by linear orders in the field and residue field sort. Secondly, we deduce from a…

We continue the analysis of definably compact groups definable in a real closed field $\mathcal{R}$. In [3], we proved that for every definably compact definably connected semialgebraic group $G$ over $\mathcal{R}$ there are a connected…

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of…

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 give a complete list of the one-dimensional groups definable in algebraically closed valued fields and i the pseudo-local fields, up to a finite index subgroup and a quotient by a finite subgroup.

We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings…

We study definable sets, groups, and fields in the theory $T_\infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an…

We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of…