On a question of Serre

Consider an imaginary quadratic number field Q(root -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 Gamma in a Bianchi group, and in particular the following question which Serre posed on p. 514 of the quoted article. Consider the map alpha induced on homology when attaching the boundary into the Borel-Serre compactification. How can one determine the kernel of alpha (in degree 1)? Serre used a global topological argument and obtained the rank of the kernel of alpha. 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 alpha into its parts associated to each cusp.

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