On pairs of finitely generated subgroups in free groups

Olshanskii, A.

doi:10.1090/proc/12537

On pairs of finitely generated subgroups in free groups
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Proc. Amer. Math. Soc. 143 (2015), 4177-4188 Request permission

Abstract:

We prove that for two arbitrary finitely generated subgroups $A$ and $B$ having infinite index in a free group $F,$ there is a subgroup $H\le B$ with finite index $[B:H]$ such that the subgroup generated by $A$ and $H$ has infinite index in $F$. The main corollary of this theorem says that a free group of free rank $r\ge 2$ admits a faithful highly transitive action, whereas the restriction of this action to any finitely generated subgroup of infinite index in $F$ has no infinite orbits.

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