Groupoids etale, eta invariants and index theory

Let G be a discrete finitely generated group. We consider a G- equivariant fibration, with fibers di¤eomorphic to a fixed even dimensional manifold with boundary Z and with base B. We assume that M is a Galois covering of a compact manifold with boundary. We consider a G-equivariant family of Dirac-type operators. Undery the assumption that the boundary family is L2 -invertible, we define an index class in the K-theory of the algebra obtained by taking the cross-product of C(B) and of G. If, in addition, G is of polynomial growth, we define higher indices by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indices: the structure of the formula is as in the seminal work of Atiyah, Patodi and Singer, with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family. Under similar assump- tions we extend our theorem to any G-proper manifold, with G an etale groupoid. We employ this generalization in order to establish a higher Atiyah-Patodi-Singer index formula on certain foliations with boundary. Fundamental to our work is a suitable generalization of Melrose b-pseudodi¤erential calculus as well as the superconnection proof of the index theorem on G-proper manifolds given by Gorokhovsky and Lott.