Highly transitive actions of groups acting on trees
HTML articles powered by AMS MathViewer
- by Pierre Fima, Soyoung Moon and Yves Stalder PDF
- Proc. Amer. Math. Soc. 143 (2015), 5083-5095 Request permission
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.References
- Gilbert Baumslag, Alexei Myasnikov, and Vladimir Remeslennikov, Malnormality is decidable in free groups, Internat. J. Algebra Comput. 9 (1999), no. 6, 687–692. MR 1727165, DOI 10.1142/S0218196799000382
- Vladimir V. Chaynikov, Properties of hyperbolic groups: Free normal subgroups, quasiconvex subgroups and actions of maximal growth, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Vanderbilt University. MR 3121970
- Yves de Cornulier, Infinite conjugacy classes in groups acting on trees, Groups Geom. Dyn. 3 (2009), no. 2, 267–277. MR 2486799, DOI 10.4171/GGD/56
- John D. Dixon, Most finitely generated permutation groups are free, Bull. London Math. Soc. 22 (1990), no. 3, 222–226. MR 1041134, DOI 10.1112/blms/22.3.222
- P. Fima, Amenable, transitive and faithful actions of groups acting on trees, arXiv:1202.6467, to appear in Annales de l’Institut Fourier.
- Shelly Garion and Yair Glasner, Highly transitive actions of $\textrm {Out}(F_n)$, Groups Geom. Dyn. 7 (2013), no. 2, 357–376. MR 3054573, DOI 10.4171/GGD/185
- A. M. W. Glass and Stephen H. McCleary, Highly transitive representations of free groups and free products, Bull. Austral. Math. Soc. 43 (1991), no. 1, 19–36. MR 1086715, DOI 10.1017/S0004972700028744
- Steven V. Gunhouse, Highly transitive representations of free products on the natural numbers, Arch. Math. (Basel) 58 (1992), no. 5, 435–443. MR 1156575, DOI 10.1007/BF01190113
- P. de la Harpe and C. Weber, Appendix by D. Osin, Malnormal subgroups and Frobenius groups: basics and examples, arXiv:1104.3065.
- K. K. Hickin, Highly transitive Jordan representations of free products, J. London Math. Soc. (2) 46 (1992), no. 1, 81–91. MR 1180884, DOI 10.1112/jlms/s2-46.1.81
- Daniel Kitroser, Highly-transitive actions of surface groups, Proc. Amer. Math. Soc. 140 (2012), no. 10, 3365–3375. MR 2929006, DOI 10.1090/S0002-9939-2012-11195-5
- T. P. McDonough, A permutation representation of a free group, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 111, 353–356. MR 453869, DOI 10.1093/qmath/28.3.353
- Soyoung Moon and Yves Stalder, Highly transitive actions of free products, Algebr. Geom. Topol. 13 (2013), no. 1, 589–607. MR 3116381, DOI 10.2140/agt.2013.13.589
- Peter M. Neumann, The structure of finitary permutation groups, Arch. Math. (Basel) 27 (1976), no. 1, 3–17. MR 401928, DOI 10.1007/BF01224634
- Jean-Pierre Serre, Arbres, amalgames, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
Additional Information
- 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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 5083-5095
- MSC (2010): Primary 20B22; Secondary 20E06, 20E08, 43A07
- DOI: https://doi.org/10.1090/proc/12659
- MathSciNet review: 3411128