Compact domination for groups definable in linear 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

Groups definable in linear o-minimal structures: the non-compact case

Let M = 〈M, +, <, 0, S〉 be a linear o-minimal expansion of an ordered group, and G = 〈G,⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ∨ definable subgroup U of 〈Mn, +〉 and a lattice L of rank equal to the dimension of the ‘compact part’ of G.

O-minimal cohomology and definably compact definable groups

LetN an o-minimal expansion of a real closed field. We develop the cohomology theory for the category ofN -definable manifolds with continuous N -definable maps and use this to solve the Peterzil-Steinhorn problem [ps] on the existence of torsion points onN -definably compact N -definable abelian groups. Namely we prove the following result: Let G be an N -definably compact N -definably connect...