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Maximal Haagerup subgroups in

Volume 275 / 2024

Alain Valette Studia Mathematica 275 (2024), 263-283 MSC: Primary 22D55; Secondary 20E25, 20E28, 20H05, 20J06 DOI: 10.4064/sm231020-15-11 Published online: 27 February 2024

Abstract

\looseness 1 For n\geq 1, let \rho _n denote the standard action of {\rm GL}_2(\mathbb {Z}) on the space P_n(\mathbb {Z})\simeq \mathbb {Z}^{n+1} of homogeneous polynomials of degree n in two variables with integer coefficients. For G a non-amenable subgroup of {\rm GL}_2(\mathbb {Z}), we describe the maximal Haagerup subgroups of the semidirect product \mathbb {Z}^{n+1}\rtimes _{\rho _n} G, extending the classification of Jiang–Skalski (2021) of the maximal Haagerup subgroups in \mathbb {Z}^2\rtimes {\rm SL}_2(\mathbb {Z}). We prove that, for n odd, the group P_n(\mathbb {Z})\rtimes {\rm SL}_2(\mathbb {Z}) has infinitely many pairwise non-conjugate maximal Haagerup subgroups which are free groups; and, for n even, P_n(\mathbb {Z})\rtimes {\rm GL}_2(\mathbb {Z}) has infinitely many pairwise non-conjugate maximal Haagerup subgroups which are isomorphic to {\rm SL}_2(\mathbb {Z}).

Authors

  • Alain ValetteInstitut de Mathématiques
    Université de Neuchâtel
    2000 Neuchâtel, Switzerland
    e-mail

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