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Ideal Uniform Polyhedra in
$\mathbb{H}^{n}$ and Covolumes of Higher Dimensional Modular Groups
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 73 / Issue 2 / April 2021
- Published online by Cambridge University Press:
- 27 January 2020, pp. 465-492
- Print publication:
- April 2021
-
- Article
- Export citation
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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 
$\unicode[STIX]{x1D701}(3)$.
