On arithmetically defined hyperbolic 5-manifolds arising from maximal orders in definite 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 $M_2\left(D\right)$ of degree 4 where $D$ denotes a definite quaternion $Q$-algebra. The affine $Z$-group scheme $S{L}_{\Lambda }$ determines an integral structure for the algebraic $Q$-group $G=S{L}_{\Lambda }{\times }_{Z}Q$ obtained by base change. The group $G$ is an inner form of the special linear $Q$-group $S{L}_4$. Each torsion-free subgroup $\Gamma \subset S{L}_{\Lambda }\left(Z\right)$ determines a hyperbolic 5-manifold, to be denoted $X_G/\Gamma$. Given a principal congruence subgroup $\Gamma \left(p^e\right)$, we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold $X_G/\Gamma \left(p^e\right)$.