Related papers: Group covers, o-minimality, and categoricity

We analyze definably compact groups in o-minimal expansions of ordered groups as a combination of semi-linear groups and groups definable in o-minimal expansions of real closed fields. The analysis involves structure theorems about their…

We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is…

We introduce a model category of spaces based on the definable sets of an o-minimal expansion of a real closed field. As a model category, it resembles the category of topological spaces, but its underlying category is a coherent topos. We…

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in…

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 Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in…

We develop in this paper the theory of covers for Hausdorff properly $\bigvee $-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an $\N$-definably connected $\N$-definable group $G$…

In this paper we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field, which we will refer to as ``definable groups''. With this terminology, it is known…

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and…

Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k>0$,…

We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive…

We sketch recent interactions between model theory and a roughly 150-year old study of analytic functions involving complex analysis, algebraic topology, and number theory, centered in canonicity of universal covers. Towards this goal we…

This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups…

By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an expansion of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show…

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various…

We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field.

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the…

An argument of A.Borel shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an…

We survey and explain some recent work at the intersection of model theory and bimeromorphic geometry (classification of compact complex manifolds). Included here are the essential saturation of the many sorted structure $\mathcal{C}$ of…