Grothendieck’s problem for 3-manifold groups

Darren D. Long; Alan W. Reid

doi:10.4171/ggd/135

JournalsggdVol. 5, No. 2pp. 479–499

Grothendieck’s problem for 3-manifold groups

Darren D. Long
University of California, Santa Barbara, USA
Alan W. Reid
University of Texas at Austin, USA

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Abstract

The following problem was posed by Grothendieck:

Let $u : H \to G$ be a homomorphism of finitely presented residually finite groups for which the extension $\overset{u}{^} : H \to G$ 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 $u : H ↪ G$ for which $\overset{u}{^}$ is an isomorphism, but $u$ is not.

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

Cite this article

Darren D. Long, Alan W. Reid, Grothendieck’s problem for 3-manifold groups. Groups Geom. Dyn. 5 (2011), no. 2, pp. 479–499

DOI 10.4171/GGD/135