[Paper Review] On the concept of fractality for groups of automorphisms of a regular rooted tree

Jone Uria–Albizuri|arXiv (Cornell University)|Apr 20, 2016

Graph theory and applications10 citations

TL;DR

This paper clarifies and distinguishes three hierarchical levels of fractality in subgroups of automorphisms of a regular rooted tree: fractal, strongly fractal, and super strongly fractal. By constructing explicit examples using GGS-groups and the Hanoi Towers group, the author proves these concepts are not equivalent—showing groups that are fractal but not strongly fractal (for d ≥ 3), strongly fractal but not super strongly fractal, and super strongly fractal groups. The key contribution is resolving ambiguity in the literature by providing definitive counterexamples and tools to analyze level stabilizer images under section homomorphisms.

ABSTRACT

The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose we define three types of fractality. We show that they are not equivalent, by giving explicit examples. Furthermore we present some tools that are helpful in order to determine the fractality of a given group.

Motivation & Objective

• To resolve confusion in the literature about the equivalence of fractal, strongly fractal, and super strongly fractal groups.
• To demonstrate that fractal does not imply strongly fractal, and strongly fractal does not imply super strongly fractal, by constructing explicit counterexamples.
• To provide tools for analyzing the image of level stabilizers under section homomorphisms ψ_u.
• To clarify the conditions under which self-similar groups exhibit recursive self-replicating properties at all levels.
• To establish that the first Grigorchuk group is super strongly fractal, not merely by implication from being strongly fractal, but through direct verification at all levels.

Proposed method

• Define three distinct fractality concepts: fractal (ψ_u(st_G(u)) = G for all u), strongly fractal (ψ_u(st_G(L_1)) = G for all u ∈ L_1), and super strongly fractal (ψ_u(st_G(L_n)) = G for all n and u ∈ L_n).
• Use GGS-groups (generalizations of the Grigorchuk and Gupta-Sidki groups) as a framework for constructing examples due to their well-understood self-similar and level-stabilizer structure.
• Apply the section homomorphism ψ_n: st_G(L_n) → G × ⋯ × G (d^n times) to analyze the image of level stabilizers.
• Leverage group presentations and commutator subgroups to estimate images of stabilizers, particularly using ψ_x(G') = ⟨b^{-1}a⟩^G to show non-surjectivity.
• Use known results on GGS-groups, such as ψ(st_G(L_n)) = st_G(L_{n-1}) × ⋯ × st_G(L_{n-1}) for n ≥ 3, to verify recursive behavior.
• Verify super strong fractality in the first Grigorchuk group by direct computation of section images at levels n = 1, 2, 3, showing all generators are realized.

Experimental results

Research questions

• RQ1 Are fractal groups necessarily strongly fractal? If not, can a counterexample be constructed for d ≥ 3?
• RQ2 Is the property of being strongly fractal equivalent to being super strongly fractal, or can a group be strongly fractal without satisfying the condition at all levels?
• RQ3 Can the image of a level stabilizer under the section homomorphism ψ_u be estimated or bounded in a way that reveals non-surjectivity?
• RQ4 Does the first Grigorchuk group satisfy super strong fractality, and if so, is this a consequence of being strongly fractal or a distinct property?
• RQ5 What conditions on the defining vector of a GGS-group ensure super strong fractality?

Key findings

• For d ≥ 3, there exist subgroups of the Hanoi Towers group that are fractal but not strongly fractal, demonstrating that fractality does not imply strong fractality.
• A GGS-group with defining vector e satisfying e_1 + ⋯ + e_{p-1} ≡ 0 mod p is super strongly fractal, providing a class of such groups.
• The first Grigorchuk group is super strongly fractal, and this is verified by direct computation of section images at levels 1, 2, and 3, not by implication from being strongly fractal.
• The group G = ⟨a,b⟩ in the Hanoi Towers subgroup is fractal but not super strongly fractal, as ψ_u(st_G(L_2)) = ⟨b^{-1}a⟩^G ≠ G, shown via the image in G/G' being cyclic.
• The image of ψ_x(st_G(L_2)) in the Hanoi Towers subgroup is ⟨[a,b],[b,a]⟩^G = G', and further analysis shows that ψ_u(st_G(L_2)) = ⟨b^{-1}a⟩^G ≠ G, proving non-super strong fractality.
• The paper establishes that strongly fractal does not imply super strongly fractal, as demonstrated by a GGS-group that is strongly fractal but fails to satisfy ψ_u(st_G(L_n)) = G for all n.