Let M =  be a linear o-minimal expansion of an ordered group, and G an n-dimensional group definable in M. 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 V-definable subgroup U of  and a lattice L of rank equal to the dimension of the 'compact part' of G. This is suggested as a structure theorem analogous to the classical theorem that every connected abelian Lie group is Lie isomorphic to a direct sum of copies of the additive group  of the reals and the circle topological group S^1. We then apply our analysis and prove Pillay's Conjecture and the Compact Domination Conjecture for a saturated M as above. En route, we show that the o-minimal fundamental group of G is isomorphic to L. Finally, we state some restrictions on L.