Isometries of CAT(0) cube complexes are semi-simple

By Frédéric Haglund. Annales mathématiques du Québec, 2007

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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.

Subfields: Geometry and Topology, Mathematical Physics

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