Denote by $\mathbb{D}$ the complete set of
reps. of cong. classes of chambers of $\nps$ obtained by the Borcherds’ method.
If $\aut_{\hbf}(D)$ is trivial for all $D\in \mathbb{D}$ then
$$ \bigcup_{D\in \mathbb{D}} D $$ is a fundamental domain of the action of $\aut(X)$ onto $\nps$.