On pairs of finitely generated subgroups in free groups
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- by A. Yu. Olshanskii PDF
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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.References
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Additional Information
- A. Yu. Olshanskii
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240 β and β Moscow State University, Moscow 119991, Russia
- MR Author ID: 196218
- Email: alexander.olshanskiy@vanderbilt.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4177-4188
- MSC (2010): Primary 20E05, 20B22, 20E07, 20E15, 54H15, 20F05
- DOI: https://doi.org/10.1090/proc/12537
- MathSciNet review: 3373918