Euler characteristics of arithmetic groups. (Q2388316)

Let \(\Gamma\) be a group. The homological Euler characteristic of \(\Gamma\) with coefficients in a representation \(V\) is defined as \[ \chi_h(\Gamma,V)=\sum_i(-1)^i\dim H^i(\Gamma,V). \] The author refines the above formula to the following: Theorem 4.1. The homological Euler characteristic of \(\Gamma\) with coefficients in \(V\) is given by \[ \chi_h(\Gamma,V)=\sum\chi(C(A))\text{Tr}(A^{-1}|V), \] where the sum is over all torsion elements of \(\Gamma\) counted up to conjugation, \(C(A)\) is the centralizer of \(A\) inside \(\Gamma\) and \(\text{Tr}(A^{-1}|V)\) is the trace of the action of \(A^{-1}\) on the finite dimensional vector space \(V\). The author applies this to obtain vanishing of \(\chi_h(\Gamma,V)\) for many subgroups of \(\text{GL}_m(K)\) commensurable to \(\text{GL}_m(\mathcal O_K)\), where \(\mathcal O_K\) is the ring of integers in a number field \(K\). The author obtains similar vanishing results for subgroups of \(\text{SL}_m(K)\) and \(\text{Sp}_m(K)\). The author also obtains the value of the Dedekind \(\zeta\) function at \(-1\) in the case of totally real number fields. This is given in terms of the ideal class set and the multiplicative group of quadratic extensions of the base ring.