# 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

## Abstract

<!-- .indented { padding-left: 50pt; padding-right: 50pt; } -->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.

## 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