Algebraic K-theory of Geometric Groups

Abstract

Let Γ be a geometrically finite group of finite asymptotic dimension and let R be a regular Noetherian ring of finite homological dimension. The main result of the paper is a proof of the Isomorphism Conjecture that the integral K-theoretic assembly map for the group ring R[Γ] is an isomorphism. We also include partial results for larger classes of geometric groups.

Excision in equivariant fibred G-theory

“…The category Fun(EΓ, C) is an interesting and useful object in its own right. There are several manifestations of this, for example in the work of Mona Merling and coauthors [14,16,17] or the work of these authors [2,4,5,7]. While in both applications it is crucial to work with the category itself, in this paper we concentrate on approximating the fixed points in Fun(EΓ, C).…”

“…A generalization of bounded K-theory has appeared in recent work on the Borel isomorphism conjecture [7,9,10,11]. The fibred homotopy fixed points defined in [10] are particularly intriguing because the standard tools in controlled algebra, based on Karoubi filtrations that generate long exact sequences, fail to compute them on a very basic level.…”

Extending properties to relatively hyperbolic groups

“…If G is torsion-free and relatively hyperbolic with respect to subgroups {H i } n i=1 satisfying the conditions of this theorem, then G also satisfies the conditions: by Corollary 3.11, G has finite decomposition complexity, and by [8,Theorem A.1], there exists a finite K(G, 1) 1 . In related work, Goldfarb [9] showed that finitely generated groups with sFDC satisfy weak regular coherence, which guarantees the existence of projective resolutions of finite length for certain R[Γ]-modules over sufficiently well-behaved coefficient rings R. This work is part of a program for proving surjectivity of assembly maps [2].…”

Topological rigidity of higher graph manifolds

“…On the other hand, relationships between several generalizations of the concept of finite asymptotic dimension in connection with isomorphism conjectures, in algebraic K and Ltheory , as well as coarse versions of these have been carried out by G. Carlsson and B. Goldfarb in [7], [12], [8].…”