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Highly transitive actions of groups acting on trees


Authors: Pierre Fima, Soyoung Moon and Yves Stalder
Journal: Proc. Amer. Math. Soc. 143 (2015), 5083-5095
MSC (2010): Primary 20B22; Secondary 20E06, 20E08, 43A07
Published electronically: August 26, 2015
MathSciNet review: 3411128
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Abstract:

We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex stabilizers and some more general, finite or infinite, edge stabilizers that we call highly core-free. We study the notion of highly core-free subgroups and give some examples. In the case of a free product amalgamated over a highly core-free subgroup and an HNN extension with a highly core-free base group we obtain a genericity result for faithful and highly transitive actions.


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Pierre Fima
Affiliation: Université Paris Diderot, Sorbonne Paris Cité, IMJ-PRG, UMR 7586, F-75013, Paris, France – and – Sorbonne Universités, UPMC Paris 06, UMR 7586, F-75013, Paris, France – and – CNRS, UMR 7586, IMJ-PRG, Case 7012, 75205 Paris, France
Email: pierre.fima@imj-prg.fr

Soyoung Moon
Affiliation: Institut Mathématiques de Bourgogne, Université de Bourgogne, CNRS UMR 5584, B.P. 47870, 21078 Dijon Cedex, France
Email: soyoung.moon@u-bourgogne.fr

Yves Stalder
Affiliation: Laboratoire de Mathématiques, Clermont Université, Université Blaise Pascal, BP 10448, F-63000 Clermont-Ferrand, France – and – CNRS UMR 6620, LM, F-63171 Aubière, France
Email: yves.stalder@math.univ-bpclermont.fr

DOI: https://doi.org/10.1090/proc/12659
Keywords: Highly transitive actions, groups acting on trees, amenable actions
Received by editor(s): November 27, 2013
Received by editor(s) in revised form: September 18, 2014
Published electronically: August 26, 2015
Additional Notes: The first author was partially supported by ANR Grants OSQPI and NEUMANN
The second author was partially supported by FABER of Conseil Régional de Bourgogne
Communicated by: Kevin Whyte
Article copyright: © Copyright 2015 American Mathematical Society