Maximal Haagerup subgroups in
Volume 275 / 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}).