This paper gives a proof of the Novikov conjecture for hyperbolic groups using the techniques of noncommutative differential geometry developed by the first author [Publ. Math., Inst. Hautes Étud. Sci. 62, 257-360 (1985; Zbl 0592.46056)].
Let be a finitely presented group and a continuous map from to . Let be the Hirzebruch -class of ; if is a class in then the number is a higher signature. The Novikov conjecture says these numbers are oriented homotopy invariants of . For the case in which is a discrete subgroup of a Lie group with finitely many components, this conjecture was proved by G. G. Kasparov using his bivariant -theory [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)]. G. G. Kasparov and G. Skandalis extended these techniques to discrete subgroups of -adic and adelic groups [C. R. Acad. Sci., Paris, Sér. I 310, No. 4, 171-174 (1990; Zbl 0705.19010)].
Let be an elliptic operator on the compact manifold . The authors show how the Alexander-Spanier cohomology of naturally pairs with to yield the localized analytic indices for . They give a cohomological formula for these localized indices using heat equation techniques based on the Getzler calculus of asymptotic pseudodifferential operators. However, the computations are far more intricate than those needed for the Atiyah-Singer index theorem.
Let be the group cohomology of , which is isomorphic to . Given , we denote by the corresponding class in . Let denote the algebraic group ring of and the algebra of smoothing operators on . Given , there is a naturally defined cyclic cocycle that gives an additive map .
We can view an elliptic operator on as a -invariant operator on the universal cover of ; thus, defines an element (this correspondence is made very explicit in the paper). If the authors show that , the localized analytic index associated with the cohomology class . In particular, when is the signature operator on , this yields the higher signature associated to .
Let be the natural inclusion and let be the induced map on -theory. Homotopy invariance of the higher signatures follows once one shows that there is a map such that . A group cocycle is called extendable if this happens for . It is here that the hyperbolic assumption on the group is used; if is hyperbolic then every group cocycle is extendable.
The authors indicate the possibility of a different approach to the Novikov theorem that avoids the localized analytic indices. There is also an aside on asymptotic cyclic cocycles and a promise of further development of this idea in a future paper.