Classifying torsion free groups in o-minimal expansions of real closed fields

In this paper we classify modulo definable group isomorphisms all torsion free groups of dimension up to four definable in an o-minimal expansion of a real closed field satisfying some conjectural assumptions. This classification implies that the number of uniformly definable families of torsion free definable groups of dimension up to four is finite and depends only on whether or not an exponential function is definable in the structure. We also adapted the Lyndon–Hochschild–Serre spectral sequence approach to the category of groups and modules definable in a fixed o-minimal structure to compute the second cohomology groups (in the o-minimal group cohomology) in the particular cases that were needed. This provides a blueprint of how to move beyond dimension four once the solvable real Lie groups are classified in higher dimensions.