Maximal Haagerup subgroups in ${\mathbb{Z}}^{n+1}{⋊}_{{\rho }_n}G{L}_2(\mathbb{Z})$

For $n\ge 1$, let ${\rho }_n$ denote the standard action of $G{L}_2(\mathbb{Z})$ 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 $G{L}_2(\mathbb{Z})$, we describe the maximal Haagerup subgroups of the semi-direct product ${\mathbb{Z}}^{n+1}{⋊}_{{\rho }_n}G$, extending the classification of Jiang-Skalski \cite{JiSk} of the maximal Haagerup subgroups in ${\mathbb{Z}}^2⋊S{L}_2(\mathbb{Z})$. We prove that, for $n$ odd, the group $P_n(\mathbb{Z})⋊S{L}_2(\mathbb{Z})$ admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are free groups; and that, for $n$ even, the group $P_n(\mathbb{Z})⋊G{L}_2(\mathbb{Z})$ admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are isomorphic to $S{L}_2(\mathbb{Z})$.