Rational embeddings of hyperbolic groups (Q2233685)

Summary: 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 \(G\).