Let G be a discrete finitely generated group. Let M M ! T be a Gequivariant fibration, with fibers di¤eomorphic to a fixed even dimensional manifold with boundary Z. We assume that G ! M M ! M M=G is a Galois covering of a compact manifold with boundary. Let À D þ ðyÞ Á y A T be a G-equivariant family of Dirac-type operators. Under the assumption that the boundary family is L 2 -invertible, we define an index class in
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 assumptions we extend our theorem to any G-proper manifold, with G an e ´tale 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 recently given by Gorokhovsky and Lott in [10]. Contents 4. Rapid decay 4.1. Virtually nilpotent groups and the rapidly decreasing algebra 4.2. The refined b-index class 5. Noncommutative di¤erential forms and higher eta invariants 5.1. Noncommutative di¤erential forms 5.2. Operators with di¤erential forms coe‰cients 5.3. The higher eta invariant 6. The b-supertrace and the higher local index theorem 6.1. The b-supertrace of an element in C Ày; d
6.2. The short time limit 7. A higher APS-index theorem: the isometric case 7.1. Isometric actions 7.2. The APS index theorem for the groupoid T z G in the isometric case 8. A higher APS index theorem: the general case 8.1. The Chern character of the index class 8.2. The Chern character of a superconnection 8.3. Main theorem and strategy of the proof 9. Proof of the main theorem 9.1. Proof of Step 3 9.2. Proof of Step 4 9.3. Proof of Step 2 10. General e ´tale groupoids 11. Applications to foliations 12. Appendix A: the rapidly decreasing b-calculus 13. Appendix B: b-smoothing operators with di¤erential form coe‰cients 14. Appendix C: a proof of theorem 3