### Abstract

Original language | English |
---|---|

Pages (from-to) | 211–221 |

Journal | Groups, Complexity, Cryptology |

Volume | 5 |

Issue number | 2 |

DOIs | |

Publication status | 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

**On the intersection of subgroups in free groups: echelon subgroups are inert.** / Rosenmann, Amnon.

Research output: Contribution to journal › Article › Research › peer-review

*Groups, Complexity, Cryptology*, vol. 5, no. 2, pp. 211–221. https://doi.org/10.1515/gcc-2013-0013

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TY - JOUR

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 -