# 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

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### Keywords

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

### Cite this

In: Groups, Complexity, Cryptology, Vol. 5, No. 2, 10.2013, p. 211–221.

Research output: Contribution to journalArticleResearchpeer-review

title = "On the intersection of subgroups in free groups: echelon subgroups are inert",
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.",
keywords = "Free groups; subgroups intersection; echelon subgroups; inert subgroups; compressed subgroups; 1-generator endomorphisms; fixed subgroups of automorphisms",
author = "Amnon Rosenmann",
year = "2013",
month = "10",
doi = "10.1515/gcc-2013-0013",
language = "English",
volume = "5",
pages = "211–221",
journal = "Groups, Complexity, Cryptology",
issn = "1869-6104",
publisher = "de Gruyter",
number = "2",

}

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T1 - On the intersection of subgroups in free groups: echelon subgroups are inert

AU - Rosenmann, Amnon

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

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

U2 - 10.1515/gcc-2013-0013

DO - 10.1515/gcc-2013-0013

M3 - Article

VL - 5

SP - 211

EP - 221

JO - Groups, Complexity, Cryptology

JF - Groups, Complexity, Cryptology

SN - 1869-6104

IS - 2

ER -