  Given a fiber bundle $Z \to M \to B$ and a flat vector bundle $E \to M$ with
a compatible action of a discrete group $G$, and regarding $B / G$ as the
non-commutative space corresponding to the crossed product algebra, we
construct an analytic torsion form as a non-commutative deRham differential
form. We show that our construction is well defined under the weaker assumption
of positive Novikov-Shubin invariant. We prove that this torsion form appears
in a transgression formula, from which a non-commutative
Riamannian-Roch-Grothendieck index formula follows.
