Centralizers in $\widetilde A_2$ groups

TL;DR

This paper investigates the structure of centralizers in torsion-free discrete groups acting on Euclidean buildings of type f0A_2, providing explicit examples and classifying their algebraic structures.

Contribution

It characterizes the centralizers in f0A_2 groups as either Bieberbach groups or finite graphs of cyclic groups, with explicit computed examples.

Findings

Centralizers are either Bieberbach groups or finite graphs of cyclic groups.

Explicit examples of centralizers in f0A_2 groups are constructed.

Provides a classification framework for centralizers in these 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 $\widetilde A_2$.