Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the K-theoretic assembly map, for groups with finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all (geometrically finite) linear groups.
Funding source: NSF grants
Award Identifier / Grant number: DMS-0804553, DMS-0968766, DMS-0706486, DMS-0600216, DMS-1101195
Funding source: ANR grants
Award Identifier / Grant number: AGORA, BLANC
We thank Daniel Kasprowski for pointing out an error in a previous version of the paper, and the referee for offering many suggestions that improved the exposition. The first author also thanks Ben Wieland for helpful conversations.
© 2014 by De Gruyter