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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The following problem was posed by Grothendieck:

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

The problem was solved in the negative by Bridson and Grunewald who produced many examples of groups GG and proper subgroups u ⁣:HGu\colon H\hookrightarrow G for which u^\hat{u} is an isomorphism, but uu 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