Virtual talk.

If an injection of finitely generated, residually finite groups $H\to G$ induces an isomorphism of profinite completions, then one says that $(G,H)$ is a Grothendieck pair. I shall describe a criterion for constructing Grothendieck pairs, emphasising the importance of finiteness properties and group homology in this context. I shall then explain how this criterion can be combined with recent advances of a different nature to produce the first examples of finitely presented, residually finite groups that are profinitely rigid in the class of finitely presented groups but not in the class of finitely generated groups. These groups are the fundamental groups of direct products of certain Seifert fibered spaces. This is joint work with Alan Reid and Ryan Spitler; it builds on our earlier work with Ben McReynolds.

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