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 $M=⟨L,R:L{R}^2=R{L}^2⟩$, 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.