Bianchi's additional symmetries

TL;DR

This paper refines the understanding of the kernel of a homology map related to Bianchi groups and their Borel-Serre compactifications, addressing a question posed by Serre with updated details.

Contribution

It provides a corrected and detailed analysis of the kernel of the homology map for Bianchi groups, building on and adjusting Serre's original topological argument.

Findings

Determined the structure of the kernel of the homology map in degree 1.

Updated Serre's original rank calculation with a precise submodule description.

Clarified the topological and algebraic properties of Bianchi groups' homology.

Abstract

In a 2012 note in Comptes Rendus Math{\'e}matique, 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 O in an imaginary quadratic number field, and the Borel-Serre compactification of the quotient of hyperbolic 3-space by SL 2 (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.