Maximal Haagerup subgroups in $\mathbb{Z}^{n+1}\rtimes_{\rho_n} {\rm GL}_2(\mathbb{Z})$
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 ${\rm SL}_2(\mathbb {Z})$.
