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

TL;DR

This paper introduces a method to compute the fixed-point proportion of certain iterated wreath products acting on regular trees, constructs a new family of self-similar groups with positive Hausdorff dimension, and explicitly calculates their fixed-point proportions.

Contribution

It develops a general method for calculating fixed-point proportions, constructs new level-transitive groups with positive Hausdorff dimension, and explicitly determines their fixed-point proportions, including for iterated Galois groups.

Findings

Explicit fixed-point proportion formulas for the new groups.

Construction of self-similar, level-transitive groups with positive Hausdorff dimension.

Application to iterated Galois groups of polynomials like x^d + 1.

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.