Rational embeddings of hyperbolic groups

Abstract

We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskiĭ. 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 $G$.