This paper studies an action of a discrete group \(\Gamma\) on a complete Riemannian manifold \(M\) with positive injectivity radius and with uniform bounds on the curvature tensor and all of its covariant derivatives. It is also assumed that \(M\) contains a “basic domain” (this is slightly more general than a fundamental domain) \(N\), such that for any proper length function \(\ell\) on \(\Gamma\), \(\ell(g)\to \infty\) in \(\Gamma\) implies that \(d(N, gN)\to \infty\) in \(M\). Here \(d\) is the Riemannian distance function. The action of \(\Gamma\) does not have to be cocompact, but it has to have only finitely many orbit types. The authors then study the maximal Roe algebra \(C^*_{\max}(M)^\Gamma\), the maximal completion of the \(*\)-algebra of \(\Gamma\)-invariant, locally compact operators with finite propagation. Under these assumptions, if the action of \(\Gamma\) is proper and isometric and if \(M\) has a \(\Gamma\)-equivariant spin structure, the authors show that the “maximal Roe index” of the Dirac operator in the \(K\)-theory of \(C^*_{\max}(M)^\Gamma\) is well-defined, and that it vanishes if \(M\) has uniformly positive scalar curvature. The hardest part of the paper involves checking that the Dirac operator is regular and essentialy self-adjoint on the relevant Hilbert module. This is not trivial even without the group action.