On Volumes of Arithmetic Quotients of SO(1,n)

Details On Volumes of Arithmetic Quotients of SO(1,n) On Volumes of Arithmetic Quotients of SO(1,n)

2003-06-30

arxiv.org/abs/math/0306423v3

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

Mathematics

Number Theory

18 pages, revised version, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4)

Scientific paper

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 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-Poincare 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.

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