Let $d$ be the hyperbolic distance on $\mathcal{H}$ and $G_0$ the subgroup of endomorphisms preserving $B$ and $\mathcal{H}$. Show that $d(gu,gv) = d(u,v)$ for all $g\in G$.
Let $d$ be the hyperbolic distance on $\mathcal{H}$ and $G_0$ the subgroup of endomorphisms preserving $B$ and $\mathcal{H}$. Show that $d(gu,gv) = d(u,v)$ for all $g\in G$.