Localization, Whitehead groups, and the Atiyah Conjecture

TL;DR

This paper investigates the algebraic structures related to the Whitehead group and division closure for certain torsionfree groups, providing new isomorphisms and properties relevant to the Atiyah Conjecture.

Contribution

It establishes an isomorphism between the weak Whitehead group and the K_1-group of the division closure for a broad class of torsionfree groups, advancing understanding of their algebraic K-theory.

Findings

Wh^w(G) is isomorphic to K_1(D(G))

D(G) is shown to be a skew field

K_1(D(G)) is the abelianization of the units in D(G)

Abstract

Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is a skew field and henc K_1(\D(G)) is the abelianization of the multiplicative group of units in D(G).