Recall that $\Gamma$ denotes the subgroup of $G_0$ formed by elements $g$ such that $g(V_\mathbb{Z})=V_\mathbb{Z}$. Show that for all $v,w\in\mathcal{H}$ and all $R\geq 0$, the set $$\{g\in\Gamma \text{ such that } d(gv,w)\leq R\}$$ is finite.
Recall that $\Gamma$ denotes the subgroup of $G_0$ formed by elements $g$ such that $g(V_\mathbb{Z})=V_\mathbb{Z}$.
Show that for all $v,w\in\mathcal{H}$ and all $R\geq 0$, the set
$$\{g\in\Gamma \text{ such that } d(gv,w)\leq R\}$$
is finite.