Profinite surface groups and the congruence kernel of arithmetic lattices in \(\text{SL}_2(\mathbb{R})\).
type: Item
instance of: scholarly article
title: Profinite surface groups and the congruence kernel of arithmetic lattices in \(\text{SL}_2(\mathbb{R})\).
zbMATH Open document ID: 1076.20020
DOI: 10.1007/BF02773529
published in: Israel Journal of Mathematics
publication date: 2005-06-09T00:00:00Z
review text: Let \(X\) be a proper, nonsingular, connected algebraic curve of genus \(g\) over the field of complex numbers. The algebraic fundamental group \(\Gamma=\pi(X)\) is the profinite completion of the fundamental group of a compact oriented 2-manifold. The aim of the paper is to analyze the subgroup structure of \(\Gamma\), proving in particular: every projective (resp. characteristic, accessible) subgroup of \(\Gamma\) is isomorphic to a normal (resp. characteristic, accessible) subgroup of a free profinite group of countable rank. The author uses this description to give a solution of the congruence subgroup problem for arithmetic lattices in \(\text{SL}_2(\mathbb{R})\): namely he proves that the congruence kernel of any such arithmetic group is isomorphic to a free profinite group of countable rank.
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 2174705
zbMATH Keywords: profinite surface groups
zbMATH Keywords: congruence subgroup problem
zbMATH Keywords: arithmetic lattices
zbMATH Keywords: projective subgroups
zbMATH Keywords: free profinite groups
zbMATH Keywords: characteristic subgroups
zbMATH Keywords: accessible subgroups
author: Pavel Zalesskij
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773529
OpenAlex ID: W1970863548
description: scientific article; zbMATH DE number 2174705