Rationality of the Gromov boundary of hyperbolic groups

In 2021, Belk, Bleak and Matucci introduced a symbolic coding of the horofunction boundary of a hyperbolic group as a clopen subset of a shift of finite type. This coding was first used by them that same year to show that hyperbolic groups embed into rational groups and then by the same authors together with Zaremsky (2023) to embed them into finitely presented simple groups (solving the Boone–Higman conjecture in the hyperbolic case). In this paper, we provide an explicit description of the Gromov boundary as a quotient of horofunction boundary based on this symbolic coding and deduce several geometric consequences. In particular, we construct a synchronous automaton which determines whether two horofunctions map to the same point in the Gromov boundary.