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

Let = ⟨ M , +, S ⟩ be a linear o-minimal expansion of an ordered group, and G = ⟨ G , ⊕, e G ) an n -dimensional group definable in . 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 ⟨ M n , +⟩ and a lattice L of rank equal to the dimension of the ‘compact part’ of G .