On the K-theory of subgroups of virtually connected Lie groups

TL;DR

This paper proves the split injectivity of the assembly map in algebraic K- and L-theory for certain subgroups of virtually connected Lie groups, supporting the Novikov conjecture for these groups.

Contribution

It establishes split injectivity of assembly maps in algebraic K- and L-theory for finitely generated subgroups of virtually connected Lie groups with finite models for proper actions.

Findings

Assembly map in algebraic K-theory is split injective for these groups.

A similar split injectivity result holds for algebraic L-theory.

The results imply the integral Novikov conjecture for these groups.

Abstract

We prove that for every finitely generated subgroup of a virtually connected Lie group which admits a finite dimensional model for the classifying space for proper actions the assembly map in algebraic K-theory is split injective. We also prove a similar statement for algebraic L-theory, which in particular implies the integral Novikov conjecture for such groups.