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SUMMARY:Detecting Free Products and Free Factors in Profinite Completions 
 - Andrei Jaikin-Zapirain (ICMAT)
DTSTART:20250728T143000Z
DTEND:20250728T153000Z
UID:TALK234262@talks.cam.ac.uk
DESCRIPTION:Let \\( \\mathcal{C} \\) be a class of groups. We say that \\t
 extbf{profinite completions detect free products} in \\( \\mathcal{C} \\) 
 if every finitely generated residually finite group \\( G \\in \\mathcal{C
 } \\)\, whose profinite completion \\( \\widehat{G} \\) decomposes as \\( 
 \\widehat{G} = L \\amalg K \\) with non-trivial \\( L \\) and \\( K \\)\, 
 also decomposes as a free product \\( G = H * U \\) with non-trivial subgr
 oups \\( H \\) and \\( U \\).\nWe also say that \\textbf{profinite complet
 ions detect free factors} in \\( \\mathcal{C} \\) if\, for every finitely 
 generated residually finite group \\( G \\in \\mathcal{C} \\) with a finit
 ely generated subgroup \\( H \\)\, such that the profinite completion \\( 
 \\widehat{G} \\) decomposes as \\( \\widehat{G} = \\overline{H} \\amalg K 
 \\)\, there exists a subgroup \\( U \\leq G \\) such that \\( G = H * U \\
 ).\nI will discuss this property in the class of fundamental groups of gra
 phs of virtually free groups with virtually cyclic edge groups. The talk i
 s based on work in progress with Henrique Souza and Pavel Zalesski.
LOCATION:Seminar Room 1\, Newton Institute
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