On pairs of finitely generated subgroups in free groups

TL;DR

The paper proves a property about finitely generated subgroups in free groups, showing how certain subgroups can be extended to maintain infinite index, and explores implications for highly transitive actions.

Contribution

It establishes a new result on the structure of finitely generated subgroups in free groups and their actions, revealing conditions for infinite index extensions and transitivity.

Findings

Existence of finite index subgroups H in B with infinite index generated by A and H

Noncyclic free groups of finite rank admit faithful highly transitive actions

Restrictions of these actions to certain subgroups have no infinite orbits

Abstract

We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary of this theorem says that a noncyclic free group of finite rank admits a faithful highly transitive action on an infinite set, whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.