Groups definable in ordered vector spaces over ordered division rings

Abstract

Let $\mathcal{M} = \langle M, +, <, 0, \{\lambda\}_{\lambda \in D}\rangle$ be an ordered vector space over an ordered division ring $D$, and $G = \langle G, \oplus, e_G \rangle$ an $n$-dimensional group definable in $\mathcal{M}. We show that if $G$ is definably compact and definably connected with respect to the $t$-topology, then it is definably isomorphic to a ‘definable quotient group’ $U/L$, for some convex $\bigvee$-definable subgroup $U$ of $\langle M^n, + \rangle$ and a lattice $L$ of rank $n$. As two consequences, we derive Pillay’s conjecture for a saturated $\mathcal{M}$ as above and we show that the o-minimal fundamental group of $G$ is isomorphic to $L$.