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Commensurability growths of algebraic groups

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Abstract

Fixing a subgroup \(\varGamma \) in a group G, the full commensurability growth function assigns to each n the cardinality of the set of subgroups \(\varDelta \) of G with \([\varGamma : \varGamma \cap \varDelta ][\varDelta : \varGamma \cap \varDelta ] \le n\). For pairs \(\varGamma \le G\), where G is a higher rank Chevalley group scheme defined over \({\mathbb {Z}}\) and \(\varGamma \) is an arithmetic lattice in G, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on G.

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Acknowledgements

We are grateful to Benson Farb and Gopal Prasad for helpful conversations. We thank Rachel Skipper for helpful comments on an earlier draft. We also thank Mathoverflow user: 41291 for informing us of [17].

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Authors and Affiliations

  1. Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY, 10031, USA

    Khalid Bou-Rabee

  2. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Tasho Kaletha

  3. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Daniel Studenmund

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  1. Khalid Bou-Rabee
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  2. Tasho Kaletha
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  3. Daniel Studenmund
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Correspondence to Daniel Studenmund.

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Khalid Bou-Rabee supported in part by National Science Foundation Grant #1405609. Tasho Kaletha supported in part by National Science Foundation Grant #1801687 and a Sloan Fellowship. Daniel Studenmund supported in part by National Science Foundation Grant #1547292.

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Bou-Rabee, K., Kaletha, T. & Studenmund, D. Commensurability growths of algebraic groups. Math. Z. 294, 1749–1757 (2020). https://doi.org/10.1007/s00209-019-02334-5

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