Centralizers in \(\tilde A_2\) groups (Q2882816)

Let \(X\) be a 2-dimensional building and \(\Gamma\) be a discrete, torsion-free, cocompact subgroup of \(\mathrm{Aut}(X)\). The object of this paper is to analyze the centralizer \(Z(g)\) in \(\Gamma\) of an element \(g\in \Gamma\). The focus is mainly on the exotic case, when there is no matrix description of \(\Gamma\) possible.NEWLINENEWLINESince \(\Gamma\) is torsion-free, any \(g\in\Gamma\) is hyperbolic and hence has an axis. The analysis of \(Z(g)\) depends on the geometric situation of this set of axis: If \(g\) has only one axis, then \(Z(g)=\langle g\rangle\) is infinite cyclic. If not, and \(g\) has a regular axis, then \(Z(g)\) is a Bieberbach group, acting discretely and cocompactly on a plane. If \(g\) has an axis which is not regular (and hence all of its axis are), then \(Z(g)/\langle g\rangle\) acts on a tree without inversion, with cyclic stabilizers.