Rational embeddings of hyperbolic groups

We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskiǐ. The proof involves assigning a system of binary addresses to points in the Gromov boundary of a hyperbolic group G, and proving that elements of G act on these addresses by asynchronous 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.