The groups in question are modeled on an abstract Wiener space. Then a group Brownian motion is defined, and its properties are studied in connection with the geometry of this group. The main results include quasi-invariance of the Gaussian (heat kernel) measure, log Sobolev inequality (following a bound on the Ricci curvature), and the Taylor isomorphism to the corresponding Fock space. The latter map is a version of the Ito-Wiener expansion in the non-commutative setting. This is a joint work with B. Driver.