Group pairs, coherence and Farrell--Jones Conjecture for $K_0$

TL;DR

This paper develops methods to analyze group pairs and their stabilisers to establish coherence properties of groups and their algebras, and proves the Farrell--Jones Conjecture for certain classes of groups.

Contribution

It introduces a new approach combining properties of group pairs to prove coherence and Farrell--Jones Conjecture results for specific group classes.

Findings

Torsion-free one-relator products of locally indicable groups are coherent if factors are coherent.

Group algebras over characteristic zero fields are coherent under certain conditions.

Extensions of hyperbolic groups by
7e are coherent.

Abstract

A group pair $(G, X)$ consists of a group $G$ together with a $G$-set $X$. Such a pair encodes properties of $G$ relative to the stabilisers of points in $X$. In this paper, we show how to combine properties of group pairs and their stabilisers to prove coherence results for $G$ and its group algebra, as well as to study the quotient of $G$ obtained by killing the stabilisers. In particular, we prove that a torsion-free one-relator product of locally indicable groups is coherent provided that both factor groups are coherent. Moreover, we show that the group algebra of such a group over a field of characteristic $0$ is coherent whenever the group algebras of the factors are coherent. As other consequences of our methods, we also show that extensions of coherent locally indicable hyperbolic groups by $\mathbb{Z}$ are coherent and that groups admitting a Cohen--Lyndon presentation satisfy the Farrell--Jones Conjecture for $K_{0}$.