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A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension

Santiago Radi

TL;DR

The paper addresses computing the fixed-point proportion for groups acting on $d$-regular trees, introducing a generalizable method for iterated wreath products and a depth-2 finite-type group construction $G_{\mathcal{Q}}^{\mathcal{P}}$ with positive Hausdorff dimension and explicit positive $ ext{FPP}$. It develops a recurrence framework via the polynomial $f_S$ to compute $ ext{FPP}(W_S)$ for iterated wreath products and then combines this with coset analysis to obtain an explicit $ ext{FPP}$ formula for $G_{\mathcal{Q}}^{\mathcal{P}}$. The paper provides two concrete constructions (Construction 1 and Construction 2) yielding positive $ ext{FPP}$ and positive Hausdorff dimension, analyzes obstructions for $d\equiv 2\pmod{4}$, and connects these groups to iterated Galois groups, notably showing $G_\infty(\mathbb{Q},x^d+1,t)$ sits inside this framework with computable $ ext{FPP}$. These results illuminate the interplay between group actions on trees, fractal dimensions, and arithmetic dynamics, delivering explicit fixed-point data in contexts previously known mostly to have zero $ ext{FPP$.

Abstract

This article provides a method to calculate the fixed-point proportion of any iterated wreath product acting on a $d$-regular tree. Moreover, the method applies to a generalization of iterated wreath products acting on a $d$-regular tree, which are not groups. As an application of this generalization, a family of groups of finite type of depth $2$ acting on a $d$-regular tree with $d \geq 3$ and $d \neq 2 \pmod{4}$ is constructed. These groups are self-similar, level-transitive, have positive Hausdorff dimension, and exhibit a positive fixed-point proportion. Unlike other groups with a positive fixed-point proportion known in the literature, the fixed-point proportion of this new family can be calculated explicitly. Furthermore, the iterated Galois group of the polynomial $x^d + 1$ with $d \geq 2$ appears in this family, so its fixed-point proportion is calculated.

A family of level-transitive groups with positive fixed-point proportion and positive Hausdorff dimension

TL;DR

The paper addresses computing the fixed-point proportion for groups acting on -regular trees, introducing a generalizable method for iterated wreath products and a depth-2 finite-type group construction with positive Hausdorff dimension and explicit positive . It develops a recurrence framework via the polynomial to compute for iterated wreath products and then combines this with coset analysis to obtain an explicit formula for . The paper provides two concrete constructions (Construction 1 and Construction 2) yielding positive and positive Hausdorff dimension, analyzes obstructions for , and connects these groups to iterated Galois groups, notably showing sits inside this framework with computable . These results illuminate the interplay between group actions on trees, fractal dimensions, and arithmetic dynamics, delivering explicit fixed-point data in contexts previously known mostly to have zero .

Abstract

This article provides a method to calculate the fixed-point proportion of any iterated wreath product acting on a -regular tree. Moreover, the method applies to a generalization of iterated wreath products acting on a -regular tree, which are not groups. As an application of this generalization, a family of groups of finite type of depth acting on a -regular tree with and is constructed. These groups are self-similar, level-transitive, have positive Hausdorff dimension, and exhibit a positive fixed-point proportion. Unlike other groups with a positive fixed-point proportion known in the literature, the fixed-point proportion of this new family can be calculated explicitly. Furthermore, the iterated Galois group of the polynomial with appears in this family, so its fixed-point proportion is calculated.
Paper Structure (15 sections, 38 theorems, 123 equations, 2 figures, 1 table)

This paper contains 15 sections, 38 theorems, 123 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. About general notation
  4. Groups acting on rooted trees
  5. Groups of finite type
  6. Fixed-point proportion
  7. Iterated Galois groups
  8. The fixed-point proportion of iterated wreath products
  9. The main construction
  10. Examples
  11. Construction 1
  12. Construction 2
  13. The problem of $d \equiv 2 \pmod{4}$
  14. Application to iterated Galois groups
  15. Open questions

Key Result

Theorem 1.1

Fix a prime power $q$ and a rational function $f \in \mathbb F_q(x)$ of degree at least $2$. Denote by $\mathop{\mathrm{Per}}\nolimits(f,\mathbb P^1(\mathbb F_{q^n}))$ the set of periodic points of $f$ in the projective space $\mathbb P^1(\mathbb F_{q^n})$. Then where $t$ transcendental over $\mathbb F_q$.

Figures (2)

  • Figure 1: Plot of the functions $f_\mathcal{P}$ for $\mathcal{P}$ subgroups of $\mathop{\mathrm{Sym}}\nolimits(3)$.
  • Figure 2: Plot of the functions $f_\mathcal{P}$ for $\mathcal{P}$ subgroups of $\mathop{\mathrm{Sym}}\nolimits(4)$.

Theorems & Definitions (62)

  • Theorem 1.1: see BridyJones2022
  • Theorem 1.2: see BridyJones2022and Juul2014
  • Theorem 1.3: see BridyJones2022 and Jones2012
  • Theorem 1.4: see AbertVirag2004
  • Theorem 1.5: see FariñaRadi2024
  • Theorem 1.6: see Jones2012
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • ...and 52 more