The Farrell-Jones Isomorphism Conjecture in K-Theory

TL;DR

This paper proves the Farrell-Jones isomorphism conjecture in algebraic K-theory for certain groups acting on trees with specific conditions on the group and coefficient ring, advancing understanding of algebraic K-theory in geometric group contexts.

Contribution

It establishes the conjecture for groups acting on trees with regular or hereditary rings, under particular conditions, extending previous results with more restricted coefficient rings.

Findings

Proves the conjecture for groups acting on trees with specific ring conditions

Identifies conditions under which the conjecture holds for these groups

Provides a weaker but significant result compared to previous broader cases

Abstract

We prove that the Farrell-Jones isomorphism conjecture for non-connective algebraic K-theory for a discrete group G and a coefficient ring R holds true if G belongs to the class of groups acting on trees, under certain conditions on G (see theorem 0.5 below) and if the coefficient ring R is either regular or hereditary, depending on the structure of G. Our result is weaker than the result that has been established in [15] which says that these groups verify the conjecture for any coefficient ring, see remark 0.6 below.