Bianchi's additional symmetries

DOI10.1007/S40062-020-00262-4zbMATH Open1530.20169arXiv2007.07604MaRDI QIDQ6345142

Alexander D. Rahm

Publication date: 15 July 2020

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.

Mathematics Subject Classification ID

Quadratic extensions (11R11) Structure of modular groups and generalizations; arithmetic groups (11F06) Fuchsian groups and their generalizations (group-theoretic aspects) (20H10)

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