Geometric properties of the golden ration Thompson's group

We show that all three golden ratio Thompson's groups $F_\tau$, $T_\tau$ and $V_\tau$ embed in the asynchronous rational group. We prove properties of the Cayley graph of the monoid $M = \langle L, R : LR^2 = RL^2 \rangle$, whose topological full group is $V_\tau$. In particular, we compute a distance function for the Cayley graph of the monoid $M$. Additionally, we prove that this Cayley graph is hyperbolic in the sense of Gromov. Our analysis reveals that the horofunction boundary of this graph is homeomorphic to a space resembling a Cantor-like set, with additional isolated points situated between each pair of breakpoints.