On a question of Serre

Alexander D. Rahm (1)

1 NUIG - Department of Mathematics

Consider the Borel-Serre compactification of the quotient of hyperbolic 3-space by a finite index subgroup in a Bianchi group, and in particular the following question which Serre posed on page 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.

Keywords: Borel-Serre compactification; Bianchi groups

Domains: K-Theory and Homology [math.KT]

Alexander D. Rahm. On a question of Serre. Comptes Rendus. Mathématique, 2012, 350 (15-16), pp.Pages 741-744. ⟨10.1016/j.crma.2012.09.001⟩. ⟨hal-00670217v4⟩