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

Publikation: Beitrag in einer FachzeitschriftArtikel

## 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.
Originalsprache englisch 211–221 Groups, Complexity, Cryptology 5 2 https://doi.org/10.1515/gcc-2013-0013 Veröffentlicht - Okt 2013