Commensurators of abelian subgroups in CAT(0) groups

We study the structure of the commensurator of a virtually abelian subgroup H in G, where G acts properly on a CAT(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CAT}(0)$$\end{document} 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 CAT(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CAT}(0)$$\end{document} 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.