Asymptotic behavior of word metrics on Coxeter groups.

Abstract

We study the geometry of tessellation defined by the walls in the Moussong complex ${\mathcal{M}}_W$ of a Coxeter group $W$. It is proved that geodesics in ${\mathcal{M}}_W$ can be approximated by geodesic galleries of the tessellation. A formula for the translation length of an element of $W$ is given. We prove that the restriction of the word metric on the $W$ to any free abelian subgroup $A$ is Hausdorff equivalent to a regular norm on $A.$