Groups with torsion, bordism and rho-invariants

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular we prove that, if Γ contains torsion and m ≡ 3 (mod 4) then M admits infinitely many different bordism classes of metrics with positive scalar curvature. This implies that there exist infinitely many concordance classes; we show that this is true even up to diffeomorphism. If Γ has certain special properties, e.g. if it contains polynomially growing conjugacy classes of finite order elements, 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 dim(M) ≡ 1 (mod 4) and Γ contains a central element of odd order, then the moduli space of metrics of positive scalar curvature (modulo the action of the diffeomorphism group) has infinitely many components, if it is not empty. Some of our invariants are the delocalized η-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).