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

TL;DR

This paper demonstrates that echelon subgroups in free groups are inert, meaning their intersection ranks with any subgroup do not exceed the subgroup's rank, expanding the class of known inert subgroups.

Contribution

It proves that echelon subgroups, including fixed subgroups of automorphisms, are inert, broadening understanding of subgroup intersections in free groups.

Findings

Echelon subgroups are inert in free groups.

Inertia holds for 1-generator endomorphisms.

Fixed subgroups of automorphisms are echelon and inert.

Abstract

A subgroup $H$ of a free group $F$ is called inert in $F$ if for every $G < F$ the rank of the intersection of $H$ with $G$ is no grater than the rank of $G$. 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.