The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert-Hadamard spaces

TL;DR

The paper proves the rational strong Novikov conjecture for countable, discrete subgroups $\Gamma$ of $\operatorname{Diff}(N)$ that are $\mu$-discrete for some regular Borel measure $\mu$ on a compact manifold $N$, extending prior volume-preserving results. It develops a comprehensive framework of infinite-dimensional geometric objects: continuous fields of Hilbert-/Hadamard spaces, their measurable and $L^2$-continuum variants, randomizations, and corresponding $C^*$-algebras, to study group actions via equivariant KK-theory. Central to the approach are deformation and trivialization techniques that convert complex actions into fiberwise-trivial data, enabling a Dirac–dual–Dirac–style analysis in the infinite-dimensional setting. The construction yields new tools for higher index theory on diffeomorphism groups and provides a pathway to deducing the Novikov conjecture, Gromov–Lawson-type scalar curvature results, and related rigidity phenomena for manifolds with such fundamental groups.

Abstract

In this paper, we prove the Novikov conjecture for a class of highly non-linear groups, namely discrete subgroups of the diffeomorphism group of a compact smooth manifold. This removes the volume-preserving condition in a previous work. This result is proved by studying operator $K$-theory and group actions on continuous fields of infinite dimensional non-positively curved spaces.

Theorems & Definitions (185)

• Theorem 1
• Theorem 2
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• Definition 2.5
• Theorem 2.6: cf. guentnerhigsontrout
• Lemma 2.9
• proof