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On pairs of finitely generated subgroups in free groups


Author: A. Yu. Olshanskii
Journal: Proc. Amer. Math. Soc. 143 (2015), 4177-4188
MSC (2010): Primary 20E05, 20B22, 20E07, 20E15, 54H15, 20F05
Published electronically: June 18, 2015
MathSciNet review: 3373918
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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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Additional Information

A. Yu. Olshanskii
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240 — and — Moscow State University, Moscow 119991, Russia
Email: alexander.olshanskiy@vanderbilt.edu

DOI: https://doi.org/10.1090/proc/12537
Keywords: Free group, coset graph, highly transitive action
Received by editor(s): August 24, 2013
Received by editor(s) in revised form: April 26, 2014
Published electronically: June 18, 2015
Additional Notes: The author was supported in part by the NSF grant DMS 1161294 and by the Russian Fund for Basic Research grant 11-01-00945
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society