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Higher torsion in the Abelianization of the full Bianchi groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 September 2013

Alexander D. Rahm
Alexander D. Rahm*
Affiliation:
Department of Mathematics, National University of Ireland at Galway,Ireland email Alexander.Rahm@nuigalway.ie

Abstract

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Denote by $ \mathbb{Q} ( \sqrt{- m} )$, with $m$ a square-free positive integer, an imaginary quadratic number field, and by ${ \mathcal{O} }_{- m} $ its ring of integers. The Bianchi groups are the groups ${\mathrm{SL} }_{2} ({ \mathcal{O} }_{- m} )$. In the literature, so far there have been no examples of $p$-torsion in the integral homology of the full Bianchi groups, for $p$ a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance $p= 80\hspace{0.167em} 737$ at the discriminant $- 1747$.

Supplementary materials are available with this article.

MSC classification

Secondary: 11F75: Cohomology of arithmetic groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 344 - 365
Copyright
© The Author(s) 2013 

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