On ideal boundaries of some Coxeter groups

Abstract

If a group acts geometrically (i.e., properly discontinuously, cocompactly and isometrically) on two geodesic spaces $X$ and $X'$, then an automorphism of the group induces a quasi-isometry $X \to X'$. We find a geometric action of a Coxeter group $W$ on a CAT(0) space $X$ and an automorphism $\phi$ of $W$ such that the quasi-isometry $X \to X$ arising from $\phi$ can not induce a homeomorphism on the boundary of $X$ as in the case of Gromov-hyperbolic spaces.