Free Factors and Profinite Completions

Author:

Garrido Alejandra12,Jaikin-Zapirain Andrei2

Affiliation:

1. Departamento de Matemáticas, Universidad Autónoma de Madrid , 28049 Madrid, Spain

2. Instituto de Ciencias Matemáticas , CSIC-UAM-UC3M-UCM, 28049 Madrid, Spain

Abstract

Abstract Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group $G$ is a free factor if and only if its closure in the profinite completion of $G$ is a profinite free factor. This generalises results by Parzanchevski and Puder for free groups that were later also proved by Wilton. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.

Funder

Spain’s Ministry of Science and Innovation

Severo Ochoa Programme for Centres of Excellence in Research and Development

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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1. Relatively projective profinite groups;European Journal of Mathematics;2023-11-20

2. The finite and solvable genus of finitely generated free and surface groups;Research in the Mathematical Sciences;2023-10-05

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