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

Abstract

Let p ≥ 3 be a prime. A generalised multi-edge spinal groupis 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 br j 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 extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multiedge 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 Z/pZ.

The profinite completion of multi-EGS groups

“…then G is said to be regular branch over K. If in the previous definition the condition |G : K| < ∞ is omitted, then G is said to be weakly regular branch over K. Lastly we note that an infinite group G is just infinite if all its proper quotients are finite, and we recall from [11,Cor. 3.5] that a torsion multi-EGS group is just infinite.…”

“…Pervova's EGSgroups were the first examples of finitely generated branch groups without the congruence subgroup property, that is, when the profinite completion of the group differs from its closure in Aut(T ); see Section 2 for definitions and details. The multi-EGS groups were first defined in [11] (though there termed generalised multi-edge spinal groups) and a certain subfamily of them was known to have profinite completion differing from the closure in the congruence topology (cf. [11,Thm.…”

“…The multi-EGS groups were first defined in [11] (though there termed generalised multi-edge spinal groups) and a certain subfamily of them was known to have profinite completion differing from the closure in the congruence topology (cf. [11,Thm. 1.4(3)]).…”

“…A multi-EGS group is a finitely generated, residually-(finite p) infinite group. It is a fractal subgroup of Aut(T ), and from [11] it is known to be just infinite when torsion. A restricted subclass of multi-EGS groups was identified in [11] as being branch; see Section 2 for relevant terminology.…”

“…It is a fractal subgroup of Aut(T ), and from [11] it is known to be just infinite when torsion. A restricted subclass of multi-EGS groups was identified in [11] as being branch; see Section 2 for relevant terminology. As we shall see, this paper identifies all multi-EGS groups that are just infinite and, respectively, branch.…”