Normal forms for the sub-Riemannian exponential map of ${\mathbb{G}}_{\alpha }$, $\mathrm{S}\mathrm{U}(2)$, and $\mathrm{S}\mathrm{L}(2)$

The goal of this paper is to use singularity theory to find normal forms near the critical points of the sub-Riemannian exponential map. Three cases are studied: the $\alpha$-Grushin plane with fold singularities, and the special unitary group $\mathrm{S}\mathrm{U}(2)$ and special linear group $\mathrm{S}\mathrm{L}(2)$ with fold and saddle-like singularities. They serve as examples of different sub-Riemannian structures and the techniques presented can be applied to other contexts. The paper also includes a discussion of the implications of this approach, as well as open problems.