Splitting With Continuous Control in Algebraic K-theory

Abstract

In this work, the continuously controlled assembly map in algebraic $K$-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups $\Gamma$ that satisfy certain geometric conditions. The group $\Gamma$ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, $K_0(k\Gamma )$ is proved to be isomorphic to the colimit of $K_0(kH)$ over the finite subgroups $H$ of $\Gamma$, when $\Gamma$ is a virtually polycyclic group and $k$ is a field of characteristic zero.

22 pages

Cited by in corpus (14)

• On the domain of the assembly map in algebraic K-theory
• On the K-theory of groups with Finite Decomposition Complexity
• Algebraic K-theory of group rings and the cyclotomic trace map
• Continuous Control and the Algebraic L-theory Assembly Map
• Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory
• Metric compactifications and coarse structures
• Induction Theorems and Isomorphism Conjectures for K- and L-Theory
• Algebraic K-theory of stable $\infty$-categories via binary complexes
• Orbit categories, classifying spaces, and generalized homotopy fixed points
• Split Injectivity of the Baum-Connes Assembly Map
• Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actions
• Controlled algebraic G-theory, I
• Assembly Maps for Group Extensions in $K$-Theory and $L$-Theory with Twisted Coefficients
• Group boundaries for semidirect products with Z