Asymptotic Behavior of Word Metrics on Coxeter Groups

We study the geometry of tessellation defined by the walls in the Moussong complex $\calm_W$ of a Coxeter group $W$. It is proved that geodesics in $\calm_W$ 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 $A.$

2010 Mathematics Subject Classification: 20F55, 51F15

Keywords and Phrases: Coxeter group, Moussong complex, translation length

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