On volumes of arithmetic quotients of \(\mathrm{SO}(1,n)\). (English) Zbl 1170.11307

Belolipetsky, Mikhail

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 3, No. 4, 749-770 (2004).

Summary: We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of \(\mathrm{SO}(1,n)\). As a result we prove that for any even dimension \(n\) there exists a unique compact arithmetic hyperbolic \(n\)-orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic 4-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic 4-manifold.

MSC:

11E57	Classical groups
22E40	Discrete subgroups of Lie groups

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