Maximal subgroups and irreducible representations of generalised multi-edge spinal groups

TL;DR

This paper studies a broad class of groups acting on p-adic trees, proving torsion groups have no infinite index maximal subgroups and constructing faithful infinite-dimensional irreducible representations.

Contribution

It extends existing techniques to a new class of generalized multi-edge spinal groups, establishing their subgroup structure and representation theory.

Findings

Torsion generalized multi-edge spinal groups have no maximal subgroups of infinite index.

Certain groups in this class admit faithful infinite-dimensional irreducible representations.

The work generalizes results from GGS-groups and Gupta-Sidki groups.

Abstract

Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised 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 and Bartholdi, to show that certain generalised multi-edge spinal groups admit faithful infinite dimensional irreducible representations over the prime field $\mathbb{Z}/p\mathbb{Z}$.