On a question of Serre

Abstract

Consider an imaginary quadratic number field Q(−m), with m a square-free positive integer, and its ring of integers O. The Bianchi groups are the groups SL2(O). Further consider the Borel–Serre compactification [7] (1970) of the quotient of hyperbolic 3-space H by a finite index subgroup Γ in a Bianchi group, and in particular the following question which Serre posed on p. 514 of the quoted article. Consider the map α induced on homology when attaching the boundary into the Borel–Serre compactification. How can one determine the kernel of α (in degree 1)? of the kernel of α. In the quoted article, Serre did add the question what submodule precisely this kernel is. Through a local topological study, we can decompose the kernel of α into its parts associated to each cusp.