Non-split sharply 2- and 3-transitive groups in SL_n(\mathbb Z)

We prove that ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$ contains a non-split sharply 2-transitive subgroup, answering a question of Glasner and Gulko. We also prove that ${\mathrm{S}\mathrm{L}}_4(\mathbb{Z})$ contains a non-split sharply 3-transitive subgroup, but that ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$ does not contain an infinite sharply 3-transitive subgroup.