Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

TL;DR

This paper proves the algebraic K-theoretic Novikov conjecture for certain groups over various operator algebras, introducing assembly maps with finite coefficients and establishing isomorphisms in algebraic K-theory for hyperbolic groups.

Contribution

It establishes the Novikov conjecture for torsion-free subgroups of linear and Lie groups over specific operator algebras and introduces assembly maps with finite coefficients.

Findings

Groups satisfy the integral algebraic K-theoretic Novikov conjecture over pt and .

Assembly maps with finite coefficients are effective under additional hypotheses.

Isomorphism between algebraic K-theory groups for hyperbolic groups and their reduced C*-algebra crossed products.

Abstract

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and \S, where \cpt denotes the C^*-algebra of compact operators and \S denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite coefficients. For all torsion free Gromov hyperbolic groups G, we demonstrate that the canonical algebra homomorphism \cpt[G]\map C^*_r(G)\hat{\otimes}\cpt induces an isomorphism between their algebraic K-theory groups.