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

Andrei Jaikin-Zapirain, Marco Linton, Pablo Sánchez-Peralta

TL;DR

The paper develops a robust framework of group pairs to attack coherence questions for groups and their group algebras, and to connect these questions to the Farrell–Jones conjecture for K0. Its core contributions include proving coherence for torsion-free one-relator products of locally indicable coherent factors, establishing coherence of group algebras in characteristic zero under similar assumptions, and obtaining surjectivity statements for K0 via Cohen–Lyndon presentations. The methods intertwine homological coherence for group pairs, L2-Betti number vanishing, and the Cohen–Lyndon property to promote from FP2-type subgroups to finitely presented subgroups, with graph-of-groups techniques handling cyclic extensions and hyperbolic contexts. The results broaden the class of groups for which K0 and coherence properties are understood, and illuminate structural links between finiteness properties, group actions on trees, and K-theoretic conjectures with implications for Wall obstructions and finite domination.

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 $K_{0}$.

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

TL;DR

The paper develops a robust framework of group pairs to attack coherence questions for groups and their group algebras, and to connect these questions to the Farrell–Jones conjecture for K0. Its core contributions include proving coherence for torsion-free one-relator products of locally indicable coherent factors, establishing coherence of group algebras in characteristic zero under similar assumptions, and obtaining surjectivity statements for K0 via Cohen–Lyndon presentations. The methods intertwine homological coherence for group pairs, L2-Betti number vanishing, and the Cohen–Lyndon property to promote from FP2-type subgroups to finitely presented subgroups, with graph-of-groups techniques handling cyclic extensions and hyperbolic contexts. The results broaden the class of groups for which K0 and coherence properties are understood, and illuminate structural links between finiteness properties, group actions on trees, and K-theoretic conjectures with implications for Wall obstructions and finite domination.

Abstract

A group pair consists of a group together with a -set . Such a pair encodes properties of relative to the stabilisers of points in . In this paper, we show how to combine properties of group pairs and their stabilisers to prove coherence results for and its group algebra, as well as to study the quotient of 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 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 are coherent and that groups admitting a Cohen--Lyndon presentation satisfy the Farrell--Jones Conjecture for .

Paper Structure

This paper contains 34 sections, 51 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Homological coherence and coherence of groups
  3. Coherence of group algebras
  4. K0 of group algebras
  5. Preliminaries
  6. General notation
  7. Finiteness Conditions on Modules
  8. One-relator products
  9. --Betti numbers of modules
  10. Graphs of groups and groups acting on trees
  11. Group pairs
  12. Group pairs and the associated augmentation module
  13. An example: 2-complexes and groups pairs
  14. The relation module
  15. Cohomological dimension of group pairs
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $A$ and $B$ be two locally indicable coherent groups and let $w\in A*B$ be an element that is not conjugated to an element in $A$ or $B$. Then the group $A*B/\langle\!\langle w \rangle\!\rangle$ is coherent.

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1
  • Theorem 1.6: WallObstruction
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • ...and 89 more