We show that an automorphism of an arbitrary CAT(0) cube complex either has a fixed point or preserves some combinatorial axis. It follows that when a group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls.
Why CAT(0) cube complexes should be replaced with median graphs
Resolutions of CAT(0) cube complexes and accessibility properties
Stallings folds for CAT(0) cube complexes and quasiconvex subgroups
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