A $C^m$ Whitney Extension Theorem for Horizontal Curves in Free Step $2$ Carnot Groups

We establish a $C^m$ Whitney extension theorem for horizontal curves in free step~$2$ Carnot groups ${\mathbb{G}}_r$ for an arbitrary number of generators $r\ge 2$. This extends existing results in the Heisenberg group. New techniques include a quantitative linear dependence structure on jets on the horizontal layer, a discretization process to correct vertical areas, and studying the effect of unrelated vertical areas.