Commensurators of abelian subgroups in CAT(0) groups

TL;DR

This paper investigates the structure of commensurators of virtually abelian subgroups in CAT(0) groups, revealing their relation to normalizers and providing applications to classifying spaces.

Contribution

It characterizes the commensurator of abelian subgroups in CAT(0) groups across different geometric settings, extending understanding of subgroup structures.

Findings

Commensurator equals the normalizer in Hadamard manifolds for semisimple subgroups.

Commensurator is an ascending union of normalizers in CAT(0) cube complexes.

Results enable construction of classifying spaces with virtually abelian stabilizers.

Abstract

We study the structure of the commensurator of a virtually abelian subgroup $H$ in $G$, where $G$ acts properly on a $\mathrm{CAT}(0)$ space $X$. When $X$ is a Hadamard manifold and $H$ is semisimple, we show that the commensurator of $H$ coincides with the normalizer of a finite index subgroup of $H$. When $X$ is a $\mathrm{CAT}(0)$ cube complex or a thick Euclidean building and the action of $G$ is cellular, we show that the commensurator of $H$ is an ascending union of normalizers of finite index subgroups of $H$. We explore several special cases where the results can be strengthened and we discuss a few examples showing the necessity of various assumptions. Finally, we present some applications to the constructions of classifying spaces with virtually abelian stabilizers.