Abstract
We consider an automorphism of an arbitrary CAT(0) cube complex. We study its combinatorial displacement and we show that either the automorphism has a fixed point or it preserves some combinatorial axis. It follows that when a f.g. group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls. As an application Baumslag-Solitar groups and Heisenberg groups provide examples of groups having a proper action on measured spaces with walls, but no proper action on a discrete space with wall.
Résumé
Nous considérons un automorphisme d’un complexe cubique CAT(0) général. Nous étudions son déplacement combinatoire, et nous établissons une dichotomie: ou bien l’automorphisme fixe un point, ou bien il préserve un axe combinatoire. Il en résulte qu’un groupe de type fini contenant un sous-groupe cyclique distordu n’agit pas proprement sur un espace à murs discret. Ainsi les groupes de Baumslag–Solitar ou de Heisenberg fournissent des exemples de groupes agissant proprement sur un espace à murs mesurés, mais pas sur un espace à murs discret.
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Haglund, F. Isometries of CAT(0) cube complexes are semi-simple. Ann. Math. Québec 47, 249–261 (2023). https://doi.org/10.1007/s40316-021-00186-2
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DOI: https://doi.org/10.1007/s40316-021-00186-2