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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
Published Online: 2013-10-11
Published in Print: 2013-11-01
© 2013 by Walter de Gruyter Berlin Boston