Torsion Free Endotrivial Modules for Finite Groups of Lie Type

In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $\mathrm{\ell }$-subgroups in any finite group of Lie type, for any prime $\mathrm{\ell }$, which may be of independent interest.