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

Joachim Schwermer

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 $Λ$ 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 $SL_Λ$ determines an integral structure for the algebraic $\mathbb{Q}$-group $G = SL_Λ \times_{\mathbb{Z}} \mathbb{Q}$ obtained by base change. The group $G$ is an inner form of the special linear $\mathbb{Q}$-group $SL_4$. Each torsion-free subgroup $Γ\subset SL_Λ(\mathbb{Z})$ determines a hyperbolic $5$-manifold, to be denoted $X_G/Γ$. Given a principal congruence subgroup $Γ(\frak{p}^e)$, we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold $X_G/Γ(\frak{p}^e)$.

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 -space we consider arithmetically defined -dimensional hyperbolic manifolds which are non-compact but of finite volume. They arise from maximal orders in the central simple algebra of degree where denotes a definite quaternion -algebra. The affine -group scheme determines an integral structure for the algebraic -group obtained by base change. The group is an inner form of the special linear -group . Each torsion-free subgroup determines a hyperbolic -manifold, to be denoted . Given a principal congruence subgroup , we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold .

Paper Structure

This paper contains 19 sections, 12 theorems, 27 equations.

Table of Contents

  1. introduction
  2. An inner form of the algebraic $\mathbb Q$-group $SL_4$
  3. Central simple algebras of degree two
  4. Ramification
  5. The group $SL_M$ and its parabolic subgroups
  6. Orders in a central simple $\mathbb Q$-algebra.
  7. Integral structures for $SL_M$ - arithmetic groups
  8. The ambient Lie group
  9. Strong approximation property
  10. Reduction theory - a suitable compactification
  11. A result in reduction theory
  12. The boundary components
  13. On the cohomology $H^*(X_G/\Gamma, \mathbb C)$
  14. The long exact sequence in cohomology attached to the pair $\overline{Y}_{\Gamma}, \partial \overline{Y}_{\Gamma}$
  15. The size of the cohomology at infinity
  16. ...and 4 more sections

Key Result

Proposition 1.1

Let $D$ be a central simple division $\mathbb Q$-algebra of degree two, and let $h_D$ be the class number of $D$. Let $G = SL_{\Lambda} \times_{\mathbb Z} \mathbb Q$ be the algebraic $\mathbb Q$-group attached to a maximal order $\Lambda = M_2(\Lambda_D)$ in the central simple $\mathbb Q$-algebra $M where $\mu_{\Lambda_D}(\frak{p}^e)$ denotes the cardinality of the finite group $\Lambda^{\times}_D

Theorems & Definitions (26)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3
  • ...and 16 more