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arXiv · 1308.3192

On pairs of finitely generated subgroups in free groups

A. Yu. Olshanskii
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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.

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A. Yu. Olshanskii. 2013-08-14. On pairs of finitely generated subgroups in free groups. https://arxiv.org/abs/1308.3192

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