Hostname: page-component-54dcc4c588-mz6gc Total loading time: 0 Render date: 2025-09-27T01:48:58.523Z Has data issue: false hasContentIssue false
  • English
  • Français

On the quasi-transitivity degree of branch groups

Part of: Structure and classification of infinite or finite groups Permutation groups

Published online by Cambridge University Press:  04 August 2025

DOMINIK FRANCOEUR
DOMINIK FRANCOEUR*
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, C/Francisco Tomás y Valiente, 7, 28049 Madrid, Spain. e-mail: dominik.francoeur@uam.es
Get access Rights & Permissions [Opens in a new window]

Abstract

An action of a group G on a set X is said to be quasi-n-transitive if the diagonal action of G on $X^n$ has only finitely many orbits. We show that branch groups, a special class of groups of automorphisms of rooted trees, cannot act quasi-2-transitively on infinite sets.

MSC classification

Primary: 20B22: Multiply transitive infinite groups
Secondary: 20E08: Groups acting on trees

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 21
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alexoudas, T., Klopsch, B. and Thillaisundaram, A.. Maximal subgroups of multi-edge spinal groups. Groups Geom. Dyn. 10(2)(2016), 619–648.10.4171/ggd/359CrossRefGoogle Scholar
Bartholdi, L., Grigorchuk, R. I. and Šuni, Z.. Branch groups. In: Handb. Algebr. vol. 3 Elsevier/North-Holland, Amsterdam, 2003, 989–1112.Google Scholar
Belk, J. and Zaremsky, M. C. B. Twisted Brin–Thompson groups. Geom. Topol. 26(3)(2022), 1189–1223.10.2140/gt.2022.26.1189CrossRefGoogle Scholar
Bondarenko, I. V.. Finite generation of iterated wreath products. Arch. Math. (Basel) 95(4)(2010), 301–308.10.1007/s00013-010-0169-2CrossRefGoogle Scholar
Cameron, P. J.. Transitivity of permutation groups on unordered sets. Math. Z. 148(2)(1976), 127–139.10.1007/BF01214702CrossRefGoogle Scholar
Cameron, P. J.. Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13(1)(1981), 1–22.10.1112/blms/13.1.1CrossRefGoogle Scholar
Cameron, P. J.. Oligomorphic Permutation Groups. London Math. Soc. Lecture Note Series, vol. 152 (Cambridge University Press, Cambridge, 1990).10.1017/CBO9780511549809CrossRefGoogle Scholar
Cameron, P. J.. Aspects of infinite permutation groups. In Groups St. Andrews 2005, vol. 1. London Math. Soc. Lecture Note Ser. vol. 339 (Cambridge University Press, Cambridge, 2007), 1–35.Google Scholar
Chaynikov, V. V.. Properties of hyperbolic groups: Free normal subgroups, quasiconvex subgroups and actions of maximal growth. (ProQuest LLC) (Ann Arbor, MI) (PhD. thesis. Vanderbilt University (2012).Google Scholar
Day, M. M.. Amenable semigroups. Illinois J. Math. 1:509–544, 1957.10.1215/ijm/1255380675CrossRefGoogle Scholar
de Cornulier, Y.. Finitely presented wreath products and double coset decompositions. Geom. Dedicata 122 (2006), 89–108.10.1007/s10711-006-9061-4CrossRefGoogle Scholar
Dixon, J. D. and Mortimer, B.. Permutation Groups. Graduate Texts in Math. vol 163 (Springer-Verlag, New York, 1996) vol. 163.10.1007/978-1-4612-0731-3CrossRefGoogle Scholar
P. Fima, F. Le Maître, Moon, S. and Stalder, Y.. A characterisation of high transitivity for groups acting on trees. Discrete Anal. pages Paper No. 8, 63, 2022.Google Scholar
Francoeur, D.. On maximal subgroups of infinite index in branch and weakly branch groups. J. Algebra 560 (2020), 818–851.10.1016/j.jalgebra.2020.06.005CrossRefGoogle Scholar
Francoeur, D. and Garrido, A.. Maximal subgroups of groups of intermediate growth. Adv. Math., 340 (2018), 1067–1107.10.1016/j.aim.2018.10.026CrossRefGoogle Scholar
Garion, S. and Glasner, Y.. Highly transitive actions of ${\rm Out}(F_n)$ . Groups Geom. Dyn. 7(2)(2013), 357–376.10.4171/ggd/185CrossRefGoogle Scholar
Glasner, Y., Souto, J. and Storm, P.. Finitely generated subgroups of lattices in ${\textrm PSL}_2\mathbb{C}$ . Proc. Amer. Math. Soc. 138(8)(2010), 2667–2676.10.1090/S0002-9939-10-10310-4CrossRefGoogle Scholar
Grigorchuk, R. I.. On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271(1)(1983), 30–33.Google Scholar
Grigorchuk, R. I.. Just infinite branch groups. In New Horizons in Pro-p Groups. Progr. Math. vol. 184 (Birkhäuser Boston, Boston, MA, 2000), 121–179.10.1007/978-1-4612-1380-2_4CrossRefGoogle Scholar
Grigorchuk, R. I.. Solved and unsolved problems around one group. In Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progr. Math. vol. 248 (Birkhäuser, Basel, 2005), 117–218.10.1007/3-7643-7447-0_5CrossRefGoogle Scholar
Hull, M. and Osin, D.. Transitivity degrees of countable groups and acylindrical hyperbolicity. Israel J. Math. 216(1)(2016), 307–353.10.1007/s11856-016-1411-9CrossRefGoogle Scholar
Klopsch, B. and Thillaisundaram, A.. Maximal subgroups and irreducible representations of generalised multi-edge spinal groups. Proc. Edinburgh. Math. Soc. (2), 61(3)(2018), 673–703.10.1017/S0013091517000451CrossRefGoogle Scholar
Le Boudec, A. and Matte Bon, N.. Triple transitivity and non-free actions in dimension one. J. London. Math. Soc. (2) 105(2)(2022), 884–908.10.1112/jlms.12521CrossRefGoogle Scholar
McDonough, T. P.. A permutation representation of a free group. Quart. J. Math. Oxford Ser. (2), 28(111)(1977), 353–356.10.1093/qmath/28.3.353CrossRefGoogle Scholar
Milnor, J.. Problem 5603. Amer. Math. Monthly 75(6)(1968), 685–686.Google Scholar
Pervova, E. L.. Everywhere dense subgroups of a group of tree automorphisms. Tr. Mat. Inst. Steklova 231 (Din. Sist., Avtom. i Beskon. Gruppy) (2000), 356–367.Google Scholar
Pervova, E. L.. Maximal subgroups of some non locally finite p-groups. Internat. J. Algebra Comput. 15(5–6)(2005), 1129–1150.10.1142/S0218196705002803CrossRefGoogle Scholar
Tsankov, T.. Automorphism groups and their actions. Thèse présentée en vue de l’obtention de l’Habilitation à Diriger des Recherches, Université Paris Diderot (2014).Google Scholar
Wilson, J. S.. Groups with every proper quotient finite. Camb. Phil. Soc. 69(1971), 373–391.10.1017/S0305004100046818CrossRefGoogle Scholar