Groups, Geometry, and Dynamics

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Volume 5, Issue 2, 2011, pp. 479–499
DOI: 10.4171/GGD/135

Published online: 2011-03-06

Grothendieck’s problem for 3-manifold groups

Darren D. Long[1] and Alan W. Reid[2]

(1) University of California, Santa Barbara, USA
(2) University of Texas at Austin, USA

The following problem was posed by Grothendieck:

Let $u \colon H\rightarrow G$ be a homomorphism of finitely presented residually finite groups for which the extension $\hat{u}\colon \widehat{H}\rightarrow \widehat{G}$ is an isomorphism. Is $u$ an isomorphism?

The problem was solved in the negative by Bridson and Grunewald who produced many examples of groups $G$ and proper subgroups $u\colon H\hookrightarrow G$ for which $\hat{u}$ is an isomorphism, but $u$ is not.

This paper addresses Grothendieck’s problem in the context of 3-manifold groups.

Keywords: Hyperbolic 3-manifold, profinite completion, character variety

Long Darren, Reid Alan: Grothendieck’s problem for 3-manifold groups. Groups Geom. Dyn. 5 (2011), 479-499. doi: 10.4171/GGD/135