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Ideal Uniform Polyhedra in $\mathbb{H}^{n}$ and Covolumes of Higher Dimensional Modular Groups

Part of: Linear algebraic groups and related topics Discontinuous groups and automorphic forms Special aspects of infinite or finite groups Zeta and $L$-functions: analytic theory Real and complex geometry

Published online by Cambridge University Press:  27 January 2020

Ruth Kellerhals
Ruth Kellerhals*
Affiliation:
Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland Email: Ruth.Kellerhals@unifr.ch
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Abstract

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.

Keywords

Ideal hyperbolic polyhedronk-rectificationNapier cycleCoxeter groupquaternionic modular grouphyperbolic volumezeta value

MSC classification

Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 20F55: Reflection and Coxeter groups 51M25: Length, area and volume
Secondary: 11F06: Structure of modular groups and generalizations; arithmetic groups 20G20: Linear algebraic groups over the reals, the complexes, the quaternions 11M06: $zeta (s)$ and $L(s, chi)$
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 465 - 492
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Copyright
© Canadian Mathematical Society 2020

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