Lusin Approximation and Rectifiability for Horizontal Curves in Carnot Groups

In Euclidean space every absolutely continuous curve admits Lusin approximations by C^1 curves. It is natural to ask whether a similar property holds in Carnot groups, where the objects of interest are horizontal curves that follow a constrained family of directions. After reviewing the positive results that are known, we describe recent work with Scott Zimmerman showing that in some Carnot groups the Lusin approximation property can fail in a remarkably strong sense. We then discuss the consequences for rectifiability of horizontal curves.