Commensurators of abelian subgroups in CAT(0) groups
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Mathematische Zeitschrift
Commensurators of abelian subgroups in CAT(0) groups Jingyin Huang1 · Tomasz Prytuła2 Received: 23 January 2019 / Accepted: 15 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We study the structure of the commensurator of a virtually abelian subgroup H in G, where G acts properly on a 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 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. Keywords Commensurator · CAT(0) group · Abelian subgroup · Hadamard manifold · CAT(0) cube complex · Bredon dimension Mathematics Subject Classification Primary 20F65; Secondary 20F67
1 Introduction Background and motivation Recall that two subgroups H1 , H2 of a group G are commensurable if H1 ∩ H2 has finite index in both H1 and H2 . The commensurator of H in G, denoted by Comm G (H ), is a subgroup consisting of all elements g ∈ G such that g H g −1 and H are commensurable. In this article we would like to understand Comm G (H ) when H is virtually abelian and G acts properly on a CAT(0) space X .
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Jingyin Huang [email protected] Tomasz Prytuła [email protected]
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Department of Mathematics, The Ohio State University, 100 Math Tower, 231 W 18th Ave, Columbus, OH 43210, USA
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Department of Applied Mathematics and Computer Science, Technical University of Denmark, Anker Engelunds Vej 1, 2800 Kgs. Lyngby, Denmark
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J. Huang, T. Prytuła
One motivation for studying such commensurators comes from the connection between some of their properties and the topology of classifying spaces of G with respect to families of virtually abelian subgroups [16]. Another motivation comes from CAT(0) geometry. For CAT(0) groups, the normalizers of their abelian subgroups are well-understood and they play a fundamental role in the theory of CAT(0) groups [2]. However, the commensurators of abelian subgroups are much more mysterious and they contain subtle information of the action which is not seen by the normalizers. The commensurator Comm G (H ) ≤ G contains normalizers of finite index subgroups of H . It is therefore natural to ask how far the commensurator is from being a normalizer. In general Comm G (H ) may not be finitely generated for a CAT(0) group G; such an example can be found in Wise’s work on irreducible lattices acting on product of trees [28], we refer to Proposition 9.1 for an explanation. On the other hand, the normalizer of H is always finitely generated [2]. Thus we ask about finitely generated subgroups of the commensurator instead. This lea
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