The Novikov conjecture, groups of diffeomorphisms, and infinite dimensional nonpositively curved spaces

Abstract: The rational strong Novikov conjecture is a prominent problem in noncommutative geometry. It implies deep conjectures in topology and differential geometry such as the (classical) Novikov conjecture on higher signatures and the Gromov-Lawson conjecture on positive scalar curvature. Using C*-algebraic and K-theoretic tools, we prove that the rational strong Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of any closed smooth manifold. The crucial geometric property of these groups that we exploit is the fact that they admit isometric and proper actions on a type of infinite-dimensional symmetric space of nonpositive curvature called the space of $L^2$-Riemannian metrics. In fact, our result holds for any discrete group admitting an isometric and proper action on a (possibly infinite-dimensional) nonpositively curved space that we call an admissible Hilbert-Hadamard space; thus our result partially extends earlier ones of Kasparov and Higson-Kasparov. This is joint work with Sherry Gong and Guoliang Yu.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides )

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