Euler characteristics of arithmetic groups

We have developed a general method for computing the homological Euler characteristic of finite index subgroups $\Gamma$ of $GL_m({\cal{O}}_K)$ where ${\cal{O}}_K$ 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 $GL_m({\cal{O}}_K)$ vanishes. For other cases, some of them very important for spaces of motivic multiple polylogarithms at $n$-th root of unity, we compute non-zero homological Euler characteristic. 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.

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Published 1 January 2005