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Commensurators of abelian subgroups in CAT(0) groups. (English) Zbl 07242434
Summary: We study the structure of the commensurator of a virtually abelian subgroup \(H\) in \(G\), where \(G\) acts properly on a \(\text{CAT}(0)\) 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 \(\text{CAT}(0)\) 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.

MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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