Groups definable in ordered vector spaces over ordered division rings

Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G,⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to t-topology, then it is definably isomorphic to a ‘definable quotient group’ U/L, for some convex W -definable subgroup U of 〈Mn, +〉 and a lattice L of rank n. As two consequences, we derive Pillay’s conjecture for M as above and we show that the o-minimal fundamental group of G is isomorphic