Profinite Subgroup Accessibility and Recognition of Amalgamated Factors

2023/08/23 by Wykowski, Julian
#20E18 20E08 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2308.12185

Abstract

We investigate accessible subgroups of a profinite group G, i.e. subgroups H appearing as vertex groups in a graph of profinite groups decomposition of G with finite edge groups. We prove that any accessible subgroup H ≤ G arises as the kernel of a continuous derivation of G in a free module over its completed group algebra. This allows us to deduce splittings of an abstract group from splittings of its profinite completion. We prove that any finitely generated subgroup Δ of a finitely generated virtually free group Γ whose closure is a factor in a profinite amalgamated product \widehatΓ = Δ \amalg_K L along a finite K must be a factor in an amalgamated product Γ= Δ∗_χΛ along some χ≅ K. This extends previous results of Parzanchevski--Puder, Wilton and Garrido--Jaikin-Zapirain on free factors.

Related

• Direct factors of profinite completions and decidability
• The profinite completion of accessible groups
• Subgroups of the direct product of graphs of groups with free abelian vertex groups
• The profinite completion of the fundamental group of infinite graphs of groups
• Profinite genus of free products with finite amalgamation