Rank-one theorem and subgraphs of BV functions in Carnot groups

We prove a rank-one theorem `a la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups ${\mathbb{H}}^n$ for $n\ge 2$. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph.