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

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 $K_{0}$.