Rational embeddings of hyperbolic groups

We prove that a large class of Gromov hyperbolic groups $G$, including all torsion-free hyperbolic groups, embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanski\u{\i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.