Pantelis Eleftheriou : Compact domination for groups definable in linear o-minimal structures

Posted by Matthias Aschenbrenner , part of the Logic Seminar.

At

Nov. 21, 2006, 4 p.m.

In

SEO 427

Abstract

We discuss Pillay's Conjecture (PC) and Compact Domination Conjecture (CDC) for groups definable in linear o-minimal structures. Let $G$ be a definably compact group definable in a saturated o-minimal structure $\mathcal{M}$. Roughly stated, PC says that $G$ must contain a normal type-definable subgroup $G^{00}$ of `infinitesimals', such that $G/G^{00}$ is a real compact Lie group of the same dimension as $G$. CDC says that, in this case, the canonical homomorphism $\pi:G\rightarrow G/G^ {00}$ is a kind of intrinsic `standard part map'. If $\mathcal{M}$ expands a real closed field, then PC is true (Hrushovski-Peterzil-Pillay) and CDC remains open. If $\mathcal{M}$ is an ordered vector space over an ordered division ring, we first prove that $G$ is a `definable torus', and then answer positively both PC and CDC.