  We develop a general method for computing the homological Euler
characteristic of finite index subgroups G of GL_m(O_K) where O_K 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 GL_m(O_K) 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 GL_3(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.
