On arithmetically defined hyperbolic 5-manifolds arising from maximal orders in definite ℚ-algebras

2024/10/22 by Joachim Schwermer, Schwermer, Joachim
Mathematics · #Geometric and Algebraic Topology #Advanced Operator Algebra Research #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2410.17107

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 M₂(D) of degree 4 where D denotes a definite quaternion ℚ-algebra. The affine ℤ-group scheme SL_Λ determines an integral structure for the algebraic ℚ-group G = SL_Λ ×_ℤ ℚ obtained by base change. The group G is an inner form of the special linear ℚ-group SL₄. Each torsion-free subgroup Γ⊂ SL_Λ(ℤ) determines a hyperbolic 5-manifold, to be denoted X_G/Γ. Given a principal congruence subgroup Γ(\frakp^e), we determine the number of ends and the dimensions of the cohomology groups at infinity of the manifold X_G/Γ(\frakp^e).

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