Euler characteristics of arithmetic groups

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We develop a general method for computing the homological Euler characteristic of finite index subgroups G of GLₘ(Oₖ) where Oₖ is the ring of integers in a number field K. With this method we find, that for large, explicitly computed dimensions m, the homological Euler characteristic of finite index subgroups of GLₘ(Oₖ) vanishes. For other cases, some of them very important for spaces of multiple polylogarithms, we compute non-zero homological Euler characteristic. With the same method we find all the torsion elements in GL₃(Z) up to conjugation. Finally, our method allows us to obtain a formula for the Dedekind zeta function at -1 in terms of the ideal class set and the multiplicative group of quadratic extensions of the base ring.