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Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathb...
This is a hard problem in general. When $N$ is not prime, then even the first homology of $\operatorname{PSL}_2(\mathbf{Z}[\frac{1}{N}])$ is non-trivial to compute (cf. Corollary 4.4 of [1]), but I re …