Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T15:01:42.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal Subgroups and Irreducible Representations of Generalized Multi-Edge Spinal Groups

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  17 April 2018

Benjamin Klopsch  and
Anitha Thillaisundaram
Benjamin Klopsch
Affiliation:
Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany ([email protected])
Anitha Thillaisundaram*
Affiliation:
School of Mathematics and Physics, University of Lincoln, Lincoln LN6 7TS, UK ([email protected])
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let p ≥ 3 be a prime. A generalized multi-edge spinal group

$$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$
is a subgroup of the automorphism group of a regular p-adic rooted tree T that is generated by one rooted automorphism a and p families $b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$ of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalizes the concepts of multi-edge spinal groups, including the widely studied GGS groups (named after Grigorchuk, Gupta and Sidki), and extended Gupta–Sidki groups that were introduced by Pervova [‘Profinite completions of some groups acting on trees, J. Algebra310 (2007), 858–879’]. Extending techniques that were developed in these more special cases, we prove: generalized multi-edge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore, we use tree enveloping algebras, which were introduced by Sidki [‘A primitive ring associated to a Burnside 3-group, J. London Math. Soc.55 (1997), 55–64’] and Bartholdi [‘Branch rings, thinned rings, tree enveloping rings, Israel J. Math.154 (2006), 93–139’], to show that certain generalized multi-edge spinal groups admit faithful infinite-dimensional irreducible representations over the prime field ℤ/pℤ.

Keywords

tree automorphismsbranch groupsmaximal subgroupsirreducible representations

MSC classification

Primary: 20E08: Groups acting on trees
Secondary: 20E28: Maximal subgroups 20C07: Group rings of infinite groups and their modules
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 673 - 703
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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.)

References

1.Alexoudas, T., Klopsch, B. and Thillaisundaram, A., Maximal subgroups of multi-edge spinal groups, Groups Geom. Dyn. 10 (2016), 619–648.Google Scholar
2.Bartholdi, L., Branch rings, thinned rings, tree enveloping rings, Israel J. Math. 154 (2006), 93–139.CrossRefGoogle Scholar
3.Bartholdi, L., Branch rings, thinned rings, tree enveloping rings – Erratum, Israel J. Math. 193 (2013), 507–508.Google Scholar
4.Bartholdi, L., Grigorchuk, R. I. and Šuniḱ, Z., Branch groups, In Handbook of algebra (ed. Hazewinkel, M.), Volume 3, pp. 989–1112 (North-Holland, Amsterdam, 2003).Google Scholar
5.Bondarenko, I. V., Finite generation of iterated wreath products, Arch. Math. (Basel) 95 (2010), 301–308.CrossRefGoogle Scholar
6.Bou-Rabee, K., Leemann, P.-H. and Nagnibeda, T., Weakly maximal subgroups in regular branch groups, J. Algebra 455 (2016), 347–357.Google Scholar
7.Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic pro-p groups (Cambridge University Press, Cambridge, 1999).Google Scholar
8.Ershov, M. and Jaikin-Zapirain, A., Groups of positive weighted deficiency and their applications, J. Reine Angew. Math. 677 (2013), 71–134.Google Scholar
9.Fernández-Alcober, G. A., Garrido, A. and Uria-Albizuri, J., On the congruence subgroup property for GGS-groups, Proc. Amer. Math. Soc. 145 (2017), 3311–3322.Google Scholar
10.Fernández-Alcober, G. A. and Zugadi-Reizabal, A., GGS-groups: order of congruence quotients and Hausdorff dimension, Trans. Amer. Math. Soc. 366 (2014), 1993–2017.CrossRefGoogle Scholar
11Francoeur, D. and Garrido, A., Maximal subgroups of groups of intermediate growth, Preprint (arXiv:1611.01216).Google Scholar
12.Garrido, A., On the congruence subgroup problem for branch groups, Israel J. Math. 216 (2016), 1–13.Google Scholar
13.Grigorchuk, R. I., Just infinite branch groups, in New horizons in pro-p groups (ed. du Sautoy, M., Segal, D. and Shalev, A.), pp. 121–179, Progress in Mathematics, Volume 184 (Birkhäuser, Boston, MA, 2000).CrossRefGoogle Scholar
14.Grigorchuk, R. I. and Wilson, J. S., The uniqueness of the actions of certain branch groups on rooted trees, Geom. Dedicata 100 (2003), 103–116.CrossRefGoogle Scholar
15.Lam, T. Y., A first course in noncommutative rings (Springer-Verlag, New York, 2001).CrossRefGoogle Scholar
16.Mazurov, V. D. and Khukhro, E. I., The Kourovka Notebook, Unsolved problems in group theory, 18th edition (Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 2016), (arXiv:1401.0300v9).Google Scholar
17.Passman, D. S., The semiprimitivity of group algebras, In Methods in ring theory (Levico Terme, 1997), Lecture Notes in Pure and Applied Mathematics, Volume 198, pp. 199–211 (Dekker, New York, 1998).Google Scholar
18.Passman, D. S., Semiprimitivity of group algebras: past results and recent progress, in Trends in Ring Theory, Proceedings of a Conference at Miskolc, Hungary 1996, CMS Conference Proceedings, Volume 22, pp. 127–157 (American Mathematical Society, Providence, RI, 1998).Google Scholar
19.Passman, D. S. and Temple, W., Representations of the Gupta–Sidki group, Proc. Amer. Math. Soc. 124 (1996), 1403–1410.Google Scholar
20.Pervova, E. L., Maximal subgroups of some non locally finite p-groups, Internat. J. Algebra Comput. 15 (2005), 1129–1150.CrossRefGoogle Scholar
21.Pervova, E. L., Profinite completions of some groups acting on trees, J. Algebra 310 (2007), 858–879.Google Scholar
22.Pervova, E. L., Profinite topologies in just infinite branch groups, Preprint 2002-154 of the Max Planck Institute for Mathematics, Bonn, Germany.Google Scholar
23.Robinson, D. J. S., A course in the theory of groups (Springer-Verlag, New York–Berlin, 1982).Google Scholar
24.Rozhkov, A. V., On the theory of groups of Aleshin type, Mat. Zametki 40 (1986), 572–589.Google Scholar
25.Sidki, S., A primitive ring associated to a Burnside 3-group, J. London Math. Soc. 55 (1997), 55–64.CrossRefGoogle Scholar
26.Smoktunowicz, A., The Jacobson radical of rings with nilpotent homogeneous elements, Bull. Lond. Math. Soc. 40(6) (2008), 917–928.Google Scholar
27.Vieira, A. C., Modular algebras of Burnside p-groups, Mat. Contemp. 21 (2001), 287–304.Google Scholar
28.Vovkivsky, T., Infinite torsion groups arising as generalizations of the second Grigorchuk group, In Algebra (Moscow, 1998) (ed. Bahturin, Y.), pp. 357–377 (de Gruyter, Berlin, 2000).Google Scholar