  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.
