On volumes of arithmetic quotients SO

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 $SO$. 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
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DOI 10.2422/2036-2145.2004.4.04
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