Continuous control and the algebraic L-theory assembly map

Abstract

Abstract. In this work, the assembly map in L-theory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by Carlsson-Pedersen to split the assembly map in the case of torsion free groups. Here, the continuously controlled techniques and results are extended to handle groups with torsion.

Integral Novikov conjectures and arithmetic groups containing torsion elements

“…In [1], the rank 0 assumption is important in order to apply the following result of Rosenthal [4,5,22].…”

On the algebraic K- and L-theory of word hyperbolic groups

“…Tools such as spectral sequences and Chern characters can then be used to calculate the homology groups, so that a piece of the geometrically important K-and Lgroups of RΓ can be understood. In this note, we show that word hyperbolic groups satisfy the conditions of [10,11], thus proving the following theorem: Theorem 1.1. Let Γ be a word hyperbolic group.…”

“…In [10,11], conditions were given for discrete groups Γ under which the assembly maps in algebraic K-and L-theory are split injective. For such groups, a portion of the K-and L-theory of a group ring RΓ is then described by an appropriate equivariant homology theory evaluated on the universal space for proper Γ-actions.…”

“…It is also important to note that in the case of torsionfree word hyperbolic groups, Theorem 1.1 follows from Carlsson and Pedersen [3]. It is proved in [10,11] that a discrete group Γ will satisfy statements (1) and (2) of Theorem 1.1 if there is a finite Γ-CW model for the universal space for proper Γ-actions that admits a compactification X such that…”

Split Injectivity of the Baum-Connes Assembly Map

“…In [CP95,Ros03a,Ros03b], the splittings were obtained by showing that the leftmost vertical map and the bottom map in the diagram were weak homotopy equivalences.…”