Detecting free factors in profinite completions

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• Alejandra Garrido (Universidad Autónoma de Madrid)
• Friday 11 November 2022, 13:45-14:45
• Zoom.

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If one is interested in a finitely generated residually finite group, it is natural to ask to what extent it is determined by its profinite completion. The extent to which this is a hard question in general is shown by the fact that it is not yet known whether a finitely generated free group is determined by its profinite completion (a question attributed to Remeslennikov). A variant of Remeslennikov’s question is whether a free factor of a group G can be detected from its profinite completion: if G has a subgroup H, whose closure in the profinite completion of G is a profinite free factor, must H be a free factor of G? This question is still hard, with positive known answer only when G itself is a free group.

 

I will report on joint work with A. Jaikin in which we provide a new proof of the above positive answer and extend it to the case when G is virtually free. The methods might be extended to other classes of groups if some interesting questions are answered on their completed group algebras.

This talk is part of the Geometric Group Theory (GGT) Seminar series.

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