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Restricted Hausdorff spectra of $q$-adic automorphisms

Jorge Fariña-Asategui

TL;DR

The paper resolves Grigorchuk's question on the self-similar Hausdorff spectrum by deriving a universal dimension formula for closed self-similar groups expressed via obstructions sequences and generating functions, and by introducing a generalized iterated wreath-product construction $G_{\mathcal{S}}$ to realize prescribed spectra. It establishes four restricted spectra for the group of $q$-adic automorphisms $\Gamma_q$: self-similar, level-transitive and super strongly fractal spectrum $[0,1]$; weakly regular branch spectrum $(0,1]$; regular branch spectrum $\mathbb{Z}[1/p]\cap(0,1]$; and self-similar branch spectrum $[0,1]$. The authors then develop a rich theory of generalized wreath products to control the sequence of obstructions $\{s_n\}$ and thereby construct self-similar and branch subgroups with prescribed Hausdorff dimensions in $\Gamma_q$, including the first examples of zero-dimensional, just infinite branch pro-$p$ groups. These results provide new insights into the interplay between fractality, branching, and Hausdorff spectra in $p$-adic automorphism groups and give strong evidence towards (and against) conjectures in the area, notably by producing zero-dimension branch examples and clarifying the spectrum structure under various finiteness and branching restrictions.

Abstract

Firstly, we completely determine the self-similar Hausdorff spectrum of the group of $q$-adic automorphisms where $q$ is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products. Secondly, we provide for every prime $p$ the first examples of just infinite branch pro-$p$ groups with zero Hausdorff dimension in $Γ_p$, giving strong evidence against a well-known conjecture of Boston.

Restricted Hausdorff spectra of $q$-adic automorphisms

TL;DR

The paper resolves Grigorchuk's question on the self-similar Hausdorff spectrum by deriving a universal dimension formula for closed self-similar groups expressed via obstructions sequences and generating functions, and by introducing a generalized iterated wreath-product construction to realize prescribed spectra. It establishes four restricted spectra for the group of -adic automorphisms : self-similar, level-transitive and super strongly fractal spectrum ; weakly regular branch spectrum ; regular branch spectrum ; and self-similar branch spectrum . The authors then develop a rich theory of generalized wreath products to control the sequence of obstructions and thereby construct self-similar and branch subgroups with prescribed Hausdorff dimensions in , including the first examples of zero-dimensional, just infinite branch pro- groups. These results provide new insights into the interplay between fractality, branching, and Hausdorff spectra in -adic automorphism groups and give strong evidence towards (and against) conjectures in the area, notably by producing zero-dimension branch examples and clarifying the spectrum structure under various finiteness and branching restrictions.

Abstract

Firstly, we completely determine the self-similar Hausdorff spectrum of the group of -adic automorphisms where is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products. Secondly, we provide for every prime the first examples of just infinite branch pro- groups with zero Hausdorff dimension in , giving strong evidence against a well-known conjecture of Boston.
Paper Structure (26 sections, 38 theorems, 91 equations, 4 figures)

This paper contains 26 sections, 38 theorems, 91 equations, 4 figures.

Table of Contents

  1. introduction
  2. Background
  3. The $m$-adic tree and its group of automorphisms
  4. Subgroups of $\mathfrak{A}$
  5. Just infinite groups
  6. Hausdorff dimension
  7. Hausdorff dimension formula and applications
  8. Definition of the sequences
  9. Motivation and main cases
  10. Proof of \ref{['theorem: Formula Hausdorff dimension']}
  11. A characterization of regular branchness
  12. Restrictions on the spectrum
  13. An application to a question of Bartholdi
  14. Interpretation of \ref{['theorem: Formula Hausdorff dimension']}
  15. A Generalization of iterated permutational wreath products
  16. ...and 11 more sections

Key Result

Theorem A

Let $\Gamma_{q}$ be the group of $q$-adic automorphisms for $q$ a $p$-power. Then the group $\Gamma_{q}$ has the following four restricted Hausdorff spectra:

Figures (4)

  • Figure 1: The definition of $r_n$ for a weakly self-similar group in terms of the subgroup $\mathfrak{S}_n$, where the same colours represent the same indices.
  • Figure 2: The definition of $r_n$ for a branching group in terms of the subgroup $\mathfrak{B}_n$, where the same colours represent the same indices.
  • Figure 3: The defining sequence $\mathcal{S}$ for the group $G_\mathcal{S}$.
  • Figure 4: The staircase part of the portrait of $b_n$.

Theorems & Definitions (73)

  • Conjecture 1: see NewHorizonsBoston
  • Theorem A
  • Theorem B
  • Proposition 1.1
  • Theorem 1.2
  • Theorem C
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 63 more