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Embracing condensed mathematics for the study of cohomological finiteness conditions of profinite groups

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Peter Kropholler
University of Southampton
Mon, 19/06/2023 - 16:30 - 17:30

A toe in the condensate: We study cohomological finiteness conditions for profinite groups by using Clausen-Scholze condensed mathematics. This approach allows one to emulate known strategies for discrete groups. We need only the most elementary constructions in condensed maths up to the notion of solidification. We outline how this approach can be used to complete the proof of a conjecture that is stated in the Ribes-Zalesski book on cohomology of profinite groups. The conjecture says every solvable pro-$p$ group of type $FP_\infty$ has finite rank and is therefore polycyclic (as a pro-$p$ group). The conjecture was proved in the torsion-free case by Ged Corob Cook and can now be proved in general. The strategy opens the way for new results about cohomological finiteness conditions for profinite groups.

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