Maximal Haagerup subgroups in $\mathbb{Z}^{n+1}\rtimes_{ρ_n}GL_2(\mathbb{Z})$

Abstract: For $n\geq 1$, let $\rho_n$ denote the standard action of $GL_2(\Z)$ on the space $P_n(\Z)\simeq\Z^{n+1}$ of homogeneous polynomials of degree $n$ in two variables, with integer coefficients. For $G$ a non-amenable subgroup of $GL_2(\Z)$, we describe the maximal Haagerup subgroups of the semi-direct product $\Z^{n+1}\rtimes_{\rho_n} G$, extending the classification of Jiang-Skalski \cite{JiSk} of the maximal Haagerup subgroups in $\Z^2\rtimes SL_2(\Z)$. We prove that, for $n$ odd, the group $P_n(\Z)\rtimes SL_2(\Z)$ admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are free groups; and that, for $n$ even, the group $P_n(\Z)\rtimes GL_2(\Z)$ admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are isomorphic to $SL_2(\Z)$.