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.
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Acknowledgements
Both authors thank the Max Planck Institute for Mathematics where part of the work was completed. Both authors thank I. Leary and A. Minasyan for valuable discussions and comments improving the paper. J. H. thanks J. Lafont, T. Nguyen and T. T. Nguyen-Phan for helpful discussions. T. P. thanks the Fields Institute for Research in Mathematical Sciences where part of the work was completed. T. P. was supported by EPSRC First Grant EP/N033787/1. T. P. thanks G. Margulis and J. Schwermer for helpful discussions.
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Huang, J., Prytuła, T. Commensurators of abelian subgroups in CAT(0) groups. Math. Z. 296, 79–98 (2020). https://doi.org/10.1007/s00209-019-02449-9
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DOI: https://doi.org/10.1007/s00209-019-02449-9