arXiv subjects
Publications and source records attributed to Frédéric Haglund.
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.
Given a complete CAT(0) space $X$ endowed with a geometric action of a group $Γ$, it is known that if $Γ$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove a converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0) realization of a Coxeter group $W$, and $Γ$ is a subgroup of $W$. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.