Progress around the Boone-Higman Conjecture

TL;DR

This paper reviews the history and recent progress on the Boone-Higman Conjecture, which links solvable word problems in finitely generated groups to embeddings into finitely presented simple groups.

Contribution

It surveys recent results that confirm the conjecture for many significant classes of groups, advancing understanding of the relationship between group properties and embeddings.

Findings

Confirmed the conjecture for several large classes of groups

Provided a historical overview of the conjecture's development

Highlighted recent techniques used in proving the conjecture

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.