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.