Torsion homology growth in arithmetic groups
type: Item
instance of: scholarly article
title: Torsion homology growth in arithmetic groups
zbMATH Open document ID: 1422.11124
DOI: 10.4171/176-1/12
author: Nicolas Bergeron
publication date: 2018-11-14T00:00:00Z
review text: This survey is rich with ideas. The author expresses some regret that various questions that could have been included, could not be covered in the article. But, the ideas, motivations to questions, known results, conjectures and approaches towards them that are sketched here are already a veritable treasure for non-experts as well as useful to experts. The results and conjectures described in the article have been obtained in collaboration with Akshay Venkatesh. Classically, only the free part of the homology groups of arithmetic groups was studied in detail. For instance, for congruence subgroups \(\Gamma_0(N)\) of \(\mathrm{SL}_2(R)\) where \(R\) is either \(\mathbb{Z}\) or the ring of integers of a real quadratic field, the free part \(H_i(\Gamma_0(N), \mathbb{Z})\) for \(i=1,2\) is intimately related to classical and Hilbert modular forms respectively. The torsion part appears to be little arithmetic interest. However, in marked contrast, when \(R\) is the ring of integers of an imaginary quadratic field, the free part is small but the torsion part is large and has deep arithmetic interest. To give a few dramatic examples due to \textit{M. H. Şengün} [Int. J. Number Theory 8, No. 2, 311--320 (2012; Zbl 1243.30085)]: When \(R = \mathbb{Z}[i]\), then \[ \Gamma_0(41+56i)^{ab} \cong \mathbb{Z}/4078793513671 \mathbb{Z} \oplus \mathbb{Z}/292306033 \mathbb{Z} \oplus \cdots \] \[ \Gamma_0(118+175i)^{ab} \cong \mathbb{Z} \oplus T \] where \(T\) is a finite abelian group of order \(>10^{310}\). As the author puts it, ``beyond the guilty pleasure of exhibiting gigantic randomly distributed primes, there are more serious reasons to study torsion in the homology''. Concerning congruence subgroups of \(\mathrm{SL}_2(R)\) for \(R\) ring of integers of an imaginary quadratic field, the following conjecture is stated: \textbf{Conjecture.} Let \(M_n \rightarrow M_0\) (\(n \in N\)) be a sequence of congruence covers of a fixed congruence hyperbolic \(3\)-manifold \(M_0\) such that…
reviewed by: Balasubramanian Sury
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 6979389
zbMATH Keywords: homology of arithmetic groups
zbMATH Keywords: torsion
zbMATH Keywords: mod-$p$ modular forms
zbMATH Keywords: $L$-functions
zbMATH Keywords: analytic torsion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/176-1/12
OpenAlex ID: W4249400819
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