Bianchi’s additional symmetries

Abstract

In a 2012 note in Comptes Rendus Mathématique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and the details of this adjustment are given in the present paper. The original question is the following. Consider the ring of integers?\(\mathcal {O}\) in an imaginary quadratic number field, and the Borel–Serre compactification of the quotient of hyperbolic 3–space by \(\mathrm {SL_2}(\mathcal {O})\). 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?\(\alpha \). He added the question what submodule precisely this kernel is.