Grothendieck’s problem for 3-manifold groups

Abstract

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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.