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Branch iterated Galois group with positive fixed-point proportion and positive Hausdroff dimension

Santiago Radi

Abstract

In this article we prove that the arithmetic profinite iterated monodromy group of a post-critically infinite unicritical polynomial is regular branch (and so of positive Hausdorff dimension), and has positive fixed-point proportion when the degree is odd. The examples are instances of a bigger family of regular branch groups constructed in this article, whose fixed-point proportion can be computed explicitly and is positive in many cases. This gives the first examples outside the binary rooted tree where a level-transitive group has positive Hausdorff dimension and positive fixed-point proportion, answering in the negative a question of Jones (2008).

Branch iterated Galois group with positive fixed-point proportion and positive Hausdroff dimension

Abstract

In this article we prove that the arithmetic profinite iterated monodromy group of a post-critically infinite unicritical polynomial is regular branch (and so of positive Hausdorff dimension), and has positive fixed-point proportion when the degree is odd. The examples are instances of a bigger family of regular branch groups constructed in this article, whose fixed-point proportion can be computed explicitly and is positive in many cases. This gives the first examples outside the binary rooted tree where a level-transitive group has positive Hausdorff dimension and positive fixed-point proportion, answering in the negative a question of Jones (2008).
Paper Structure (15 sections, 28 theorems, 126 equations, 2 figures, 1 table)

This paper contains 15 sections, 28 theorems, 126 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

Let $K$ be a field and $K^{sep}$ a separable closure of $K$. Let $t$ be a transcendental element over $K$, $d \geq 2$, $c \in K$ and $\xi \in K^{sep}$ a primitive $d$th root of unity. Suppose that $f(x) = x^d + c \in K[x]$ is a post-critically infinite unicritical polynomial. Then, the group $G_\inf and its fixed-point proportion is where $I$ is the image of the map $\mathop{\mathrm{Gal}}\nolimit

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 (57)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5: Sunic2006
  • ...and 47 more