Integral Novikov conjectures and arithmetic groups containing torsion elements

摘要

In this paper, we study a generalized integral Novikov conjecture for discrete groups containing nontrivial torsion elements and prove it for not necessarily torsion-free arithmetic groups of reductive algebraic groups defined over Q and virtually polycyclic groups. For this purpose, we prove a general criterion that this generalized integral Novikov conjecture holds for groups Gamma having finite asymptotic dimension and satisfying suitable conditions related to actions by finite subgroups on the universal space E-F Gamma for proper actions. For arithmetic groups Gamma, we show that the Borel-Serre partial compactification (X) over bar (BS) is a Gamma-cofinite universal space for proper actions, which is of interest independent of the application in this paper, and satisfies these other conditions as well. For virtually polycyclic groups Gamma, we use the filtration induced from the canonical decreasing commuting series to understand the structure of fixed-point sets on a model E-F Gamma given by homogeneous spaces.