Linear Groups Definable in o-Minimal Structures

Compact domination for groups definable in linear o-minimal structures

Groups Definable in O-minimal Structures

In this series of lectures, we will a) introduce the basics of ominimality, b) describe the manifold topology of groups definable in o-minimal structures, and c) present a structure theorem for the special case of semi-linear groups, exemplifying their relation with real Lie groups. The structure of these lectures is as follows: (1) Basics of o-minimality: definition, Cell Decomposition Theorem...

Solvable groups definable in o-minimal structures

Let N be an o-minimal structure. In this paper we develop group extension theory over N and use it to describe N -definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal’cev theorem and we describe N -definable G-modules and N -definable

Type-definable and invariant groups in o-minimal structures

Let M be a big o-minimal structure and G a type-definable group in Mn. We show that G is a type-definable subset of a definable manifold in Mn that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some Mk with the topology induced by Mk. Part of this result holds for th...

Higher Homotopy of Groups Definable in O-minimal Structures

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their unive...