On arithmetically defined hyperbolic $5$-manifolds arising from maximal orders in definite $\mathbb{Q}$-algebras

Abstract

Using the quaternionic formalism for the description of the group of isometries of hyperbolic $5$-space we consider arithmetically defined $5$-dimensional hyperbolic manifolds which are non-compact but of finite volume. They arise from maximal orders $\Lambda$ in the central simple algebra $M_2(D)$ of degree $4$ where $D$ denotes a definite quaternion $\mathbb{Q}$-algebra. The affine $\mathbb{Z}$-group scheme $SL_{\Lambda}$ determines an integral structure for the algebraic $\mathbb{Q}$-group $G = SL_{\Lambda} \times_{\mathbb{Z}} \mathbb{Q}$ obtained by base change. The group $G$ is an inner form of the special linear $\mathbb{Q}$-group $SL_4$. Each torsion-free subgroup $\Gamma \subset SL_{\Lambda}(\mathbb{Z})$ determines a hyperbolic $5$-manifold, to be denoted $X_G/\Gamma$. Given a principal congruence subgroup $\Gamma(\frak{p}^e)$, we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold $X_G/\Gamma(\frak{p}^e)$.