Uniqueness for the Signature of a Path of Bounded Variation and Continuous Analogues for the Free Group

This paper is at an interface between analysis, topology and control theory. In it we prove a non-commutative analogue of the result that functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. The equivalence relation determined by the “null sets” for this new problem enable us to construct a continuous analogue of the free group as a quotient subspace of the space of paths of bounded variation. In this development our main theorem extends the work of K.T. Chen, in particular the paper [1], where he considers a similar theorem under a greater smoothness assumption.