  A geometric p-rough path can be seen to be a genuine path of finite
p-variation with values in a Lie group equipped with a natural distance. The
group and its distance lift (R^{d},+,0) and its Euclidean distance. This
approach allows us to easily get a precise modulus of continuity for the
Enhanced Brownian Motion (the Brownian Motion and its Levy Area). As a first
application, extending an idea due to Millet & Sanz-Sole, we characterize the
support of the Enhanced Brownian Motion (without relying on correlation
inequalities). Secondly, we prove Schilder's theorem for this Enhanced Brownian
Motion. As all results apply in Hoelder (and stronger) topologies, this extends
recent work by Ledoux, Qian, Zhang (2002). Lyons' fine estimates in terms of
control functions allow us to show that the Ito map is still continuous in the
topologies we introduced. This provides new and simplified proofs of the
Stroock-Varadhan support theorem and the Freidlin-Wentzell theory. It also
provides a short proof of modulus of continuity for diffusion processes along
old results by Baldi.
