Progress around the Boone-Higman Conjecture

TL;DR

The Boone–Higman conjecture asks whether solvability of the word problem for a finitely generated group $G$ exactly corresponds to $G$ embedding into a finitely presented simple group. The surveyed work traces the conjecture from Higman’s embedding theorem to modern container strategies, detailing how families like Thompson groups, Röver–Nekrashevych groups, and twisted Brin–Thompson groups provide natural hosts for such embeddings. A key highlight is the hyperbolic case, where every hyperbolic group embeds into a finitely presented simple group via horofunction boundaries and contracting rational similarity groups, illustrating the power of geometric and self-similar constructions. The compilation also records substantial progress for broad classes (e.g., RAAGs, Coxeter groups, hyperbolic groups) and notes important open problems, charting a path toward a more complete Boone–Higman picture. Overall, the work advances our understanding of how algorithmic properties of groups interface with the rich structure of finitely presented simple groups, with potential practical implications for group-theoretic decision problems and embedding theory.

Abstract

A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group $G$ has solvable word problem if and only if $G$ can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

Theorems & Definitions (51)

• Proposition 2.1
• proof
• Remark 2.2
• Remark 2.3
• Corollary 2.4
• proof
• Remark 2.5
• Corollary 2.6
• proof
• Corollary 2.8