The integral homology of $PSL_2$ of imaginary quadratic integers with non-trivial class group

Alexander Rahm (1, 2) , Mathias Fuchs (3)

1 Mathematisches Institut der Universitaet Goettingen
2 IF - Institut Fourier
3 Georg-August-University = Georg-August-Universität Göttingen

We show that a cellular complex described by Floege allows to determine the integral homology of the Bianchi groups $PSL_2(O_{-m})$, where $O_{-m}$ is the ring of integers of an imaginary quadratic number field $\rationals[\sqrt{-m}]$ for a square-free natural number $m$. We use this to compute in the cases m = 5, 6, 10, 13 and 15 with non-trivial class group the integral homology of $PSL_2(O_{-m})$, which before was known only in the cases m = 1, 2, 3, 7 and 11 with trivial class group.

Mots clés: Cohomology of arithmetic groups; Discrete subgroups of Lie groups; Discontinuous groups of transformations

Domaines: K-théorie et homologie [math.KT]

Alexander Rahm, Mathias Fuchs. The integral homology of $PSL_2$ of imaginary quadratic integers with non-trivial class group. Journal of Pure and Applied Algebra, 2011, 215, pp.1443-1472. ⟨10.1016/j.jpaa.2010.09.005⟩. ⟨hal-00370722v12⟩