Embedding hyperbolic groups into finitely presented infinite simple groups

TL;DR

This paper explores the embedding of hyperbolic groups into finitely presented simple groups, advancing the understanding of the Boone--Higman conjecture and related group theory structures.

Contribution

It discusses the history of the Boone--Higman conjecture and highlights recent work showing hyperbolic groups embed into finitely presented simple groups.

Findings

Hyperbolic groups embed into finitely presented simple groups

Overview of classes of finitely presented simple groups

Progress towards the Boone--Higman conjecture

Abstract

The Boone--Higman conjecture is that every recursively presented group with solvable word problem embeds in a finitely presented simple group. We discuss a brief history of this conjecture and work towards it. Along the way we describe some classes of finitely presented simple groups, and we briefly outline work of Belk, Bleak, Matucci, and Zaremsky showing that the broad class of hyperbolic groups embeds in a class of finitely presented simple groups.