Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups

Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which $\Bir(X)$ is infinite and the Kawamata-Morrison movable cone conjecture is satisfied. For this family, the movable cone is explicitly described; it's fractal nature is related to limit sets of Kleinian groups and to the Apollonian Gasket. Then, we produce explicit examples of (biregular) automorphisms with positive entropy on some Calabi-Yau manifolds.
Comment: 30 pages. Though the most main theorem is the same, many explanations and results are polished up or made stronger. Theorem 4.6 (based on section 2.2) is not in the previous version. A version with figure is in http://perso.univ-rennes1.fr/serge.cantat/