On torsion in the cohomology of \(\operatorname{PSL}_2 (\mathbb Z[1/N])\)
Let \(p\) be a prime and let \(\Gamma\) be the \(S\)-arithmetic group \(\text{PSL}_2 (\mathbb{Z}[\frac 1p])\). Let \(X\) be the symmetric space attached to \(\Gamma\) which is a product of the upper half plane and a Bruhat-Tits tree. Let \(\partial\Gamma \setminus X\) be the boundary of the Borel-Serre compactification of \(\Gamma \setminus X\). Torsion in the group cohomology of \(\Gamma\) is known to be linked to the cohomology of \(\partial\Gamma\setminus X\). In this paper explicit calculations are done for the cohomology of \(\partial\Gamma\setminus X\) with coefficients in the Eichler-Shimura modules \(M_n\) over a field \(K\). If the characteristic of \(K\) is large enough, the author gives a list of the dimensions of the cohomology groups. To derive these, he uses an explicit description of the relevant Borel-Serre compactifications.
scientific article; zbMATH DE number 1915506
scientific article