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

TL;DR

This paper studies 5-dimensional hyperbolic manifolds derived from maximal orders in certain quaternionic algebras, analyzing their geometric and cohomological properties using arithmetic and algebraic methods.

Contribution

It introduces a new framework for constructing and analyzing hyperbolic 5-manifolds from maximal orders in definite quaternion algebras, detailing their ends and cohomology.

Findings

Determined the number of ends of the manifolds.

Computed the dimensions of cohomology groups at infinity.

Analyzed the structure of manifolds arising from principal congruence subgroups.

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 $X_G/\Gamma(\frak{p}^e)$.