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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 (klopsch@math.uni-duesseldorf.de)
Anitha Thillaisundaram*
Affiliation:
School of Mathematics and Physics, University of Lincoln, Lincoln LN6 7TS, UK (anitha.t@cantab.net)
*
*Corresponding author.
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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 

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