Euler characteristics of arithmetic groups

Abstract

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.