Related papers: Solvable groups definable in o-minimal structures

Let $L/K$ be a finite Galois extension of fields with group $\Gamma$. Associated to each Hopf-Galois structure on $L/K$ is a group $G$ of the same order as the Galois group $\Gamma$. The type of the Hopf-Galois structure is by definition…

The following refinement of the Higman embedding theorem is proved: A finitely generated group $R$ is recursively presented if and only if there exists a quasi-isometric malnormal embedding of $R$ into a finitely presented group $H$ such…

The Whitney group $K_1(A)$ is isomorphic to $A^\times \times \operatorname{SK}_1(A)$ for some subgroup $\operatorname{SK}_1(A)$, where $A$ is a commutative ring and $A^\times$ denotes the set of units in $A$. Consider an o-minimal expansion…

In this article, we give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of Lie groups defined by T.Robart [13], we define the closed holonomy group of a…

Let $G$ be a finite group and denote by $o(g)$ the order of an element $g\in G$. We say that $G$ is an $LCM$-group if $o(x^ny)$ is a divisor of the least common multiple of $o(x^n)$ and $o(y)$ for all $x, y\in G$ and $n\in\mathbb{N}$. This…

We discuss definable compactifications and topological dynamics. For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where…

In this paper we study analogues of amenability for topological groups in the context of definable structures. We prove fixed point theorems for such groups. More importantly, we propose definitions for definable actions and continuous…

Let $ M_0 $ denote either the field structure $ \mathbb{Q}_p $ of $ p $-adic numbers, or an $o$-minimal expansion of the field structure $ \mathbb{R} $ of real numbers. We investigate the minimal flows and Ellis groups of definable groups…

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$…

We show that for any finitely generated group G, group cohomology classes represented by cocycles of subexponential growth are extendable over the topological K-groups of the Lafforgue algebra associated to G.

In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups,…

We study the second cohomology group with coefficients in the adjoint module for a class of solvable Lie algebras $\mathcal{R}_{\mathcal{T}}$ that arise as maximal solvable extensions of nilpotent Lie algebras $\mathcal{N}$ of maximal rank.…

For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is…

We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and a finite number of definable subsets of its realization |K| in R we prove that there exists a triangulation (K',f) of |K| compatible with…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

This paper answers several open questions around structures with o-minimal open core. We construct an expansion of an o-minimal structure $\mathcal{R}$ by a unary predicate such that its open core is a proper o-minimal expansion of…

We construct a simplified resolution for the trivial G-module Z, where G is a finite abelian group, and compare it with the standard resolution. We use it to calculate cohomologies of irreducible G-lattices and their duals.

A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish…

We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie…

We prove a structure theorem for periodic locally soluble groups satisfying a chain condition on intersections of relatively uniformly definable subgroups using results from the theory of stable groups. The result in particular shows that…