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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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On the intersection of subgroups in free groups: Echelon subgroups are inert

Amnon Rosenmann
Published Online: 2013-10-11 | DOI: https://doi.org/10.1515/gcc-2013-0013


A subgroup H of a free group F is called inert in F if for every . In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of finitely generated free groups are echelon subgroups. The proofs follow mostly a graph-theoretic or combinatorial approach.

Keywords: Free groups; subgroups intersection; echelon subgroups; inert subgroups; compressed subgroups; 1-generator endomorphisms; fixed subgroups of automorphisms

About the article

Received: 2013-01-30

Revised: 2013-05-25

Published Online: 2013-10-11

Published in Print: 2013-11-01

Citation Information: Groups - Complexity - Cryptology, Volume 5, Issue 2, Pages 211–221, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2013-0013.

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© 2013 by Walter de Gruyter Berlin Boston.Get Permission

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