Hyperbolic groups satisfy the Boone-Higman conjecture

TL;DR

This paper proves that hyperbolic groups satisfy the Boone-Higman conjecture by embedding them into finitely presented simple groups using a new class of groups called rational similarity groups.

Contribution

It introduces rational similarity groups and demonstrates their role in embedding hyperbolic groups into finitely presented simple groups, confirming the conjecture for this class.

Findings

Hyperbolic groups embed in finitely presented simple groups.

Rational similarity groups are useful in group embedding problems.

Contracting self-similar groups satisfy the Boone-Higman conjecture.

Abstract

The 1973 Boone-Higman conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this paper, we show that hyperbolic groups satisfy this conjecture, that is, each hyperbolic group embeds in some finitely presented simple group. This shows that the conjecture holds in the "generic" case for finitely presented groups. Our key tool is a new family of groups, which we call "rational similarity groups (RSGs)", that is interesting in its own right. We prove that every hyperbolic group embeds in a full, contracting RSG, and every full, contracting RSG embeds in a finitely presented simple group, thus establishing the result. Another consequence of our work is that all contracting self-similar groups satisfy the Boone-Higman conjecture.