On the Farrell-Jones Conjecture and its applications

Abstract

Abstract. We present the status of the Farrell-Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true and study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture and conjectures generalizing Moody's Induction Theorem. Thus we extend the class of groups for which these conjectures are known considerably. Mathematics Subject Classification (2000). 19Dxx, 19A31,19B28Keywords. Algebraic K-theory of group rings with arbitrary coefficients, Farrell-Jones Conjecture, Bass Conjecture, Kaplansky Conjecture, Moody's Induction Theorem. Introduction and statements of results0.1. Background. The Farrell-Jones Conjecture for algebraic K-theory predicts the structure of K n (RG) for a group G and a ring R. There is also an L-theory version. For applications in topology and geometry the case R = Z is the most important one since many topological invariants of manifolds and CW -complexes such as the finiteness obstruction, the Whitehead torsion and the surgery obstruction take values in the algebraic K-or L-theory of the integral group ring Zπ of the fundamental group π. The Farrell-Jones Conjecture for R = Z implies several famous conjectures, e.g., the Novikov Conjecture, (in high dimensions) the Borel Conjecture, and the triviality of compact h-cobordisms with torsionfree fundamental group. On the other hand proofs of the Farrell-Jones Conjecture for certain groups often rely on working with integral coefficients since they are based on these geometric connections. This is the reason why more is known about the algebraic K-and L-theory of ZG than of CG which is in some sense surprising since CG has better ring theoretic properties than ZG. For the status of the Farrell-Jones Conjecture with coefficient in Z we refer for instance to [45, Sections 5.2 and 5.3].Recently the geometric approaches have been generalized so far that they also apply to other coefficient rings than Z (see for instance , BartelsReich [8], Bartels-Lück-Reich [6], Quinn [54]). This is interesting for algebraic and ring theoretic applications, where one would like to consider for example fields, rings of integers in algebraic number fields and integral domains. The purpose of

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

“…This implies that the algebraic K-theory groups of their group rings may be computed from the algebraic K-theory groups of their virtually cyclic subgroups via the assembly maps. More information on these topics may be found in [3], [8] and [17]. The main results of this paper were subsequently used by Millán-Ló pez in her thesis [20] where she calculated the lower algebraic K-theory of the group rings of P n ðS 2 Þ and P n ðRP 2 Þ.…”

Assembly maps

“…Here are some consequences of the Farrell-Jones Conjecture. For more information about these and other applications we refer for instance to [9,75,78]. 7.2.1.…”

A short note about diffuse Bieberbach groups

“…For more information about these objects we refer the reader to [1], [9,Chapter 10] and [7]. In part 3 of [7] the authors prove that any torsion-free crystallographic group (Bieberbach group) with trivial center is not diffuse.…”

“…The interest in diffuse groups follows from Bowditch's observation that they have the unique product property 1 . Originally unique products were introduced in the study of group rings of discrete, torsion-free groups.…”