Etale Groupoids, eta invariants and index theory

Let $\mathrm{\Gamma }$ be a discrete finitely generated group. Let $\widehat{M}\to T$ be a $\mathrm{\Gamma }$-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary $Z$. We assume that $\Gamma\to \hat{M}\to \hat{M}/\Gamma$isaGaloiscoveringofacompactmanifoldwithboundary\mathrm{.}Let$(D^+ (\theta))_{\theta\in T}$bea$\Gamma$-equivariant family of Dirac-type operators. Under the assumption that the boundary family is $L^2$-invertible, we define an index class in the K-theory of the cross-product algebra, $K_0 (C^0 (T)\rtimes_r \Gamma)$\mathrm{.}If,inaddition,$\Gamma$ is of polynomial growth, we define higher indeces by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indeces: 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 assumptions 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 $b$-pseudodifferential calculus as well as the superconnection proof of the index theorem on $G$-proper manifolds recently given by Gorokhovsky and Lott.