Logic Seminar
Pantelis Eleftheriou
University of Notre Dame
Compact domination for groups definable in linear o-minimal structures
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.
Tuesday November 21, 2006 at 4:00 PM in SEO 427