Peterzil-Steinhorn subgroups of Real Algebraic Groups

We consider Peterzil-Steinhorn groups defined in o-minimal expansions of the reals. These are one-dimensional torsion-free subgroups of arbitrary definable and not definably compact groups. We show that each Peterzil-Steinhorn group is isomorphic to either the additive or the multiplicative group of the reals and we provide a simple criterion that can be used to classify each such group into one of those two categories. Additionally, we find the tangent space of any arbitrary Peterzil-Steinhorn group at its identity. Finally, for the case of polynomially bounded o-minimal expansions of the reals, we give a complete description of all Peterzil-Steinhorn groups.