  In the context of controlled differential equations, the signature is the
exponential function on paths. B. Hambly and T. Lyons proved that the signature
of a bounded variation path is trivial if and only if the path is tree-like. We
extend Hambly-Lyons' result and their notion of tree-like paths to the setting
of weakly geometric rough paths in a Banach space. At the heart of our approach
is a new definition for reduced path and a lemma identifying the reduced path
group with the space of signatures.
