  Let G be a discrete group, and let M be a closed spin manifold of dimension
m>3 with pi_1(M)=G. We assume that M admits a Riemannian metric of positive
scalar curvature. We discuss how to use the L2-rho invariant and the
delocalized eta invariant associated to the Dirac operator on M in order to get
information about the space of metrics with positive scalar curvatur1e.
  In particular we prove that, if G contains torsion and M is congruent 3 mod 4
then M admits infinitely many different bordism classes of metrics with
positive scalar curvature. We show that this is true even up to diffeomorphism.
  If G has certain special properties then we obtain more refined information
about the ``size'' of the space of metric of positive scalar curvature, and
these results also apply if the dimension is congruent to 1 mod 4. For example,
if G contains a central element of odd order, then the moduli space of metrics
of positive scalar curvature has infinitely many components, if it is not
empty.
  Some of our invariants are the delocalized eta-invariants introduced by John
Lott. These invariants are defined by certain integrals whose convergence is
not clear in general, and we show, in effect, that examples exist where this
integral definitely does not converge, thus answering a question of Lott.
  We also discuss the possible values of the rho invariants of the Dirac
operator and show that there are certain global restrictions (provided the
scalar curvature is positive).
