 Weak geometric p-rough path

A weak geometric $p$-rough path is a continuous path in the free step-N nilpotent Lie group, with $N=\lfloor p\rfloor$, whose increments have finite $p$-variation for the Carnot-Carathéodory distance. Its multiplicative increments are ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$. They obey the Chen identity and the algebraic symmetries of path signatures.