Conjugacy classes of polyspinal groups

TL;DR

This paper investigates conjugacy conditions in spinal and multi-GGS groups, introduces polyspinal groups as a common generalization, and addresses a question about the isomorphism of finitely generated branch groups.

Contribution

It provides necessary and sufficient conditions for conjugacy in multi-GGS groups and introduces polyspinal groups, expanding understanding of branch group classifications.

Findings

Necessary condition for spinal groups to be conjugate

Necessary and sufficient condition for multi-GGS groups to be conjugate

Negative answer to the question on isomorphism of finitely generated branch groups

Abstract

Spinal groups and multi-GGS groups are both generalisations of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we give a necessary condition for spinal groups to be conjugate, and we establish a necessary and sufficient condition for multi-GGS groups to be conjugate. We also introduce a natural common generalisation of both classes, which we call polyspinal groups. Our results enable us to give a negative answer to a question of Bartholdi, Grigorchuk and Sunik, on whether every finitely generated branch group is isomorphic to a weakly branch spinal group.