Grothendieck's problem for 3-manifold groups
type: Item
instance of: scholarly article
title: Grothendieck's problem for 3-manifold groups
zbMATH Open document ID: 1250.57027
DOI: 10.4171/GGD/135
author: D. D. Long
author: Alan W. Reid
published in: Groups, Geometry, and Dynamics
publication date: 2012-02-29T00:00:00Z
review text: The problem referred to in the title of the paper is the following: Let \(u: H \rightarrow G\) be a homomorphism of finitely presented residually finite groups for which the extension \(\hat{u}: \hat{H} \rightarrow \hat{G}\) between profinite completions is an isomorphism. Is \(u\) an isomorphism? The authors introduce a notion of Grothendieck rigidity as follows: Let \(G\) be a group and \(H < G\). \((G, H)\) is a Grothendieck Pair if the inclusion of \(H\) in \(G\) provides a negative answer to Grothendieck's problem above. If for all finitely generated subgroups \(H < G\), \((G, H)\) is not a Grothendieck Pair then \(G\) is said to be Grothendieck rigid. The authors prove that any closed 3-manifold admitting a geometric structure is Grothendieck rigid. They also show that any finite volume, cusped hyperbolic 3-manifold is Grothendieck rigid.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 6010067
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: profinite completion
zbMATH Keywords: character variety
reviewed by: Mahan Mj
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6010067