# On the intersection of subgroups in free groups: echelon subgroups are inert

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## Abstract

A subgroup $H$ of a free group $F$ is called inert in $F$ if $rk(H \cap G) \leq rk(G)$ for every $G < F$. 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.
Original language English 211–221 Groups, Complexity, Cryptology 5 2 https://doi.org/10.1515/gcc-2013-0013 Published - Oct 2013

## Keywords

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