Centralizers in $\widetilde A_2$ groups

Abstract

Let $\Gamma$ be a torsion free discrete group acting cocompactly on a two dimensional euclidean building $\Delta$. The centralizer of an element of $\Gamma$ is either a Bieberbach group or is described by a finite graph of finite cyclic groups. Explicit examples are computed, with $\Delta$ of type $\widetilde A_2$.