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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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