Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Boone--Higman Conjecture for groups acting on locally finite trees

Kai-Uwe Bux, Claudio Llosa Isenrich, Xiaolei Wu

TL;DR

The paper develops a method to verify the Boone–Higman Conjecture for groups acting on locally finite trees by embedding them into finitely presented simple groups. It constructs rigid permutation groups $\mathrm{RP}_G(\mathcal{T})$ and a Stein–Farley cube complex $\mathfrak{X}$ with a height function to apply Bestvina–Brady Morse theory, proving finiteness properties via highly connected descending links. The approach, combined with Belk–Zaremsky’s twisted Brin–Thompson framework and Bass–Serre theory, yields embeddings into simple groups of type $F_n$ for broad classes: Baumslag–Solitar groups, free-by-cyclic groups, Leary–Minasyan groups, Euclidean triangle Artin groups, and the generalised BS groups $\mathcal{BS}_G$. This provides a unified route to BH for many action-on-tree classes and opens avenues for extending to further families through graph-of-groups techniques. The results significantly broaden the catalog of groups known to satisfy the Boone–Higman Conjecture and connect algorithmic properties to deep geometric and combinatorial structures.

Abstract

We develop a method for proving the Boone--Higman Conjecture for groups acting on locally finite trees. As a consequence, we prove the Boone--Higman Conjecture for all Baumslag--Solitar groups and for all free(finite rank)-by-cyclic groups, solving it in two cases that have been raised explicitly by Belk, Bleak, Matucci and Zaremsky. We also illustrate that our method has applications beyond these cases and may offer a route for proving the Boone--Higman Conjecture for many classes of groups.

On the Boone--Higman Conjecture for groups acting on locally finite trees

TL;DR

The paper develops a method to verify the Boone–Higman Conjecture for groups acting on locally finite trees by embedding them into finitely presented simple groups. It constructs rigid permutation groups and a Stein–Farley cube complex with a height function to apply Bestvina–Brady Morse theory, proving finiteness properties via highly connected descending links. The approach, combined with Belk–Zaremsky’s twisted Brin–Thompson framework and Bass–Serre theory, yields embeddings into simple groups of type for broad classes: Baumslag–Solitar groups, free-by-cyclic groups, Leary–Minasyan groups, Euclidean triangle Artin groups, and the generalised BS groups . This provides a unified route to BH for many action-on-tree classes and opens avenues for extending to further families through graph-of-groups techniques. The results significantly broaden the catalog of groups known to satisfy the Boone–Higman Conjecture and connect algorithmic properties to deep geometric and combinatorial structures.

Abstract

We develop a method for proving the Boone--Higman Conjecture for groups acting on locally finite trees. As a consequence, we prove the Boone--Higman Conjecture for all Baumslag--Solitar groups and for all free(finite rank)-by-cyclic groups, solving it in two cases that have been raised explicitly by Belk, Bleak, Matucci and Zaremsky. We also illustrate that our method has applications beyond these cases and may offer a route for proving the Boone--Higman Conjecture for many classes of groups.
Paper Structure (19 sections, 44 theorems, 18 equations, 5 figures)

This paper contains 19 sections, 44 theorems, 18 equations, 5 figures.

Table of Contents

  1. Rigid permutation groups
  2. A connectivity lemma
  3. A Stein--Farley complex for rigid permutation groups
  4. Leaf expansions and the history of trees
  5. Bestvina--Brady Morse theory
  6. Analysis of the descending links
  7. Descending links in $\mathfrak{X}$
  8. Connectivity of descending links
  9. Analysis of the cell stabilisers
  10. Oligomorphic actions
  11. Main Theorem about actions on trees
  12. Application to graphs of groups
  13. Generalised Baumslag--Solitar groups
  14. Applications to explicit examples
  15. Torsion-free strongly scale-invariant groups
  16. ...and 4 more sections

Key Result

Theorem A

The Boone--Higman Conjecture holds for the following classes of groups:

Figures (5)

  • Figure 3.1: A finite graph $\Gamma$ equipped with an admissible system of gates indicated by the orange half-edges. The walls of the half-spaces illustrate the directions in which the gates are "closed".
  • Figure 3.2: (1) The finite graph $\Gamma$ from Figure \ref{['fig:system-of-gates-for-a-finite-graph']} equipped with the admissible system of gates $\mathfrak{G}$ indicated in orange; (2) an admissible tree for the universal cover $\mathcal{T}:=\widetilde{\Gamma}$ of the finite graph $\Gamma$ that can serve as base tree $T_0$; (3) a finite subtree of $\mathcal{T}$ which contains $T_0$ and which is not admissible with respect to $\mathfrak{G}$; (4) an admissible tree obtained as an elementary expansion of $T_0$ in two leaves, with the carets we attached to $T_0$ highlighted in yellow and turquoise. Note that in the picture when there are more than one gates between a pair of adjacent vertices, we use letters in $\{a,b\}$ to distinguish them.
  • Figure 6.1: Construction of the tree $T'$ in the proof of Lemma \ref{['lem:connectivity-of-descending-links']} with Steps (1), (2) and (3) marked in red, purple and orange.
  • Figure 6.2: The tree $R$ constructed in the proof of Lemma \ref{['lem:connectivity-of-descending-links']}. We indicate the reducible carets from $\rho$ (resp. $\sigma'$) in green (resp. magenta).
  • Figure 10.1: Examples of carets in an augmented graph of groups

Theorems & Definitions (97)

  • Conjecture : Boone--Higman
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Example 1.5
  • Remark 1.6
  • ...and 87 more