Related papers: Groups and rings definable in d-minimal structures

We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor $s$ from a category $G$ to $\Top^{\op}$ and show that it defines what we call a skew category algebra $A…

Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information…

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

We study definably amenable groups in NIP theories, and answer a question of Newelski (and also of Chernikov-Simon), by giving an example in the o-minimal context where weak generic types do not coincide with almost periodic types,…

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many…

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial…

We introduce almost cohomology groups for Lie rings definable in finite-dimensional theory. In particular, we define the 0th and 1st almost cohomology groups of a Lie ring module. Moreover, we prove that the 1st almost cohomology group of a…

We continue our earlier investigation of dp-finite fields. We show that the "heavy sets" of [6] are exactly the sets of full dp-rank. As a consequence, full dp-rank is a definable property in definable families of sets. If $I$ is the group…

We answer in the affirmative a conjecture of Berarducci, Peterzil and Pillay \cite{BPP10} for solvable groups, which is an o-minimal version of a particular case of Milnor's isomorphism conjecture \cite{jM83}. We prove that every abstract…

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a…

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely…

We give a characterization of limits of dihedral groups in the space of finitely generated marked groups. We also describe the topological closure of dihedral groups in the space of marked groups on a fixed number of generators.

Every bounded definable open set is a union of finitely many open strong cells in a weakly o-minimal expansion of a real closed field. We prove this fact and another theorem similar to it.

We give examples of $\mathrm{NIP}$ structures in which new algebraic structure appears in the Shelah completion. In particular we construct a weakly o-minimal structure $\mathscr{M}$ such that $\mathscr{M}$ does not interpret an infinite…

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as Andr\'e-Oort conjecture. Among the many tools developed in…

We study various notions of "tameness" for definably complete expansions of ordered fields. We mainly study structures with locally o-minimal open core, d-minimal structures, and dense pairs of d-minimal structures.

We show how the framework of crossed simplicial groups may be used to provide a classification of topological field theories on open cobordism categories defined by reductions of the structure group to a planar Lie group. Such theories are…

We prove the triviality of the Grothendieck ring of a integer-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K^2 to itself minus a point. When we specialize to…

A visceral structure on M is given by a definable base for a uniform topology on its universe in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. This context includes o-minimal ordered…

We consider definable topological dynamics for $NIP$ groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable…