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Branch iterated Galois groups with positive fixed-point proportion and positive Hausdorff dimension

This paper establishes that the arithmetic profinite iterated monodromy groups of post-critically infinite unicritical polynomials are regular branch groups with positive Hausdorff dimension and positive fixed-point proportion (for odd degrees), providing the first examples outside the binary rooted tree that answer Jones's 2008 question in the negative.

Original authors: Santiago Radi

Published 2026-08-12

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CC BY 4.0

Original authors: Santiago Radi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world built not of atoms, but of choices. In the branch of mathematics known as group theory, scientists study "symmetry" and "structure" by looking at how objects can be shuffled around without breaking their rules. One particularly fascinating playground for this is the "rooted tree." Picture a family tree that never ends: it starts with a single ancestor at the top, and every person has exactly the same number of children, who in turn have the same number of children, forever. This is a "regular rooted tree."

Now, imagine a group of "automorphisms"—think of them as magical shufflers. These shufflers can rearrange the branches of the tree, but they must follow strict rules: they can't break the connections, and they can't change the number of children anyone has. Some of these shufflers are incredibly powerful and can move any branch to any other branch at the same level; mathematicians call these "level-transitive" groups.

For decades, mathematicians have been trying to measure two specific things about these shufflers. First, how "big" or "dense" are they inside the universe of all possible shufflers? This is measured by something called "Hausdorff dimension." Think of it like checking how much space a cloud of smoke fills in a room; a dimension of 1 means it fills the room, while 0 means it's just a thin, invisible thread. Second, how often do these shufflers leave at least one branch exactly where it started? This is the "fixed-point proportion." If you shuffle a deck of cards, a "fixed point" is a card that stays in its original spot. A high proportion means the shuffler leaves many things alone; a low proportion means it's a chaotic mover that changes almost everything.

For a long time, a big question hung over this field: Is it possible to find a group that is both "big" (filling a lot of space, having positive Hausdorff dimension) and "leaves many things alone" (leaving a significant number of branches alone, having a positive fixed-point proportion)? A famous mathematician, Jones, guessed that the answer was no. He thought that if a group was big enough to be interesting, it would have to be so chaotic that it would never leave anything alone. This paper sets out to test that guess.

The author, Santiago Radi, proves that Jones was wrong. The paper constructs specific examples of these magical shufflers that are both "big" and "leave many things alone." The main discovery is a new family of groups built from a special type of polynomial equation (specifically, equations like $x^d + c$ ). When you look at the symmetries of the solutions to these equations, you get a group that acts on the infinite tree. Radi shows that if the degree of the equation (the number $d$ ) is an odd number, this group has a positive Hausdorff dimension (it's a substantial cloud of smoke) and a positive fixed-point proportion (it leaves a measurable number of branches untouched).

To understand how this works, imagine the tree as a giant game of "choose your own adventure." The author builds a group where the rules for shuffling are very specific. At every step down the tree, the shuffler must choose a move from a specific set of allowed patterns. The paper introduces a clever way to calculate exactly how often these shufflers leave a branch alone. It turns out that for these specific groups, you can count the "leaves many things alone" moves using a simple formula involving prime numbers. If the number of branches at each step ( $d$ ) is odd, the math guarantees that there is a non-zero chance the shuffler will leave something in place.

The paper also tackles a related question about "branch groups." These are groups that, deep down in the tree, act like smaller copies of themselves. The author proves that these specific groups are "regular branch," which is a fancy way of saying they have a very structured, self-repeating nature that guarantees they are "big" (positive Hausdorff dimension). By combining this structural proof with the calculation of the "leaves many things alone" moves, the paper delivers a definitive counterexample to Jones's conjecture.

It is important to note that the paper doesn't just guess or simulate these results; it provides a rigorous mathematical proof. The author constructs the groups explicitly, defines the rules for their actions, and uses algebraic formulas to calculate the exact proportions. The paper also explores what happens when the number of branches is even. In those cases, the math suggests the "leaves many things alone" might disappear (the fixed-point proportion could be zero), but for the odd cases, the proof is solid.

So, what does this mean for the curious teenager? It means that in the vast, infinite forest of mathematical symmetries, there are indeed "big" groups that aren't as chaotic as we thought. They can fill up a lot of space while still being gentle enough to leave some things exactly where they started. This discovery opens the door to new ways of understanding how numbers and shapes interact in the deep, hidden layers of mathematics, proving that sometimes, the biggest things can still be the most careful.

Technical Summary: Branch Iterated Galois Groups with Positive Fixed-Point Proportion and Positive Hausdorff Dimension

Problem Statement
The paper addresses a specific question in arithmetic dynamics and the theory of groups acting on rooted trees, originally posed by Jones (2008). The central problem concerns the relationship between the Hausdorff dimension and the fixed-point proportion (FPP) of level-transitive subgroups of the automorphism group of a $d$ -regular rooted tree, $\text{Aut}(T)$ .

Jones conjectured that if a level-transitive group $G \leq \text{Aut}(T)$ has a positive Hausdorff dimension ( $H(G) > 0$ ), then its fixed-point proportion must be zero ($FPP(G) = 0$). While Boston (2008) provided a counterexample to this conjecture on the binary tree ( $d=2$ ), no examples were known for trees of degree $d > 2$ , nor were there examples of such groups arising as iterated Galois groups (arithmetic iterated monodromy groups). The paper specifically investigates whether regular branch iterated Galois groups can simultaneously possess positive Hausdorff dimension and positive fixed-point proportion.

Methodology
The author employs a combination of group-theoretic construction and arithmetic dynamics analysis:

Generalization of Iterated Wreath Products: The paper extends the definition of iterated wreath products $W_S$ to arbitrary subsets $S \subseteq \text{Sym}(d)$ . It establishes a recurrence relation for the fixed-point proportion of these sets, deriving a characteristic polynomial $f_S(x)$ . The FPP is shown to be the largest fixed point of this polynomial in $[0, 1]$ .
Construction of Regular Branch Groups ( $G^P_Q$ ): A new family of groups, denoted $G^P_Q$ , is constructed for subgroups $Q \triangleleft P \leq \text{Sym}(d)$ . These groups consist of automorphisms where the action on the children of any vertex lies in the same coset of $P/Q$ . The paper proves these groups are regular branch (and thus have positive Hausdorff dimension) if $Q$ is transitive.
Explicit Calculation of FPP: Using the characteristic polynomial method, the author derives an explicit formula for the FPP of $G^P_Q$ as an average of the FPPs of the iterated wreath products associated with the cosets of $P/Q$ .
Arithmetic Realization: The constructed groups are linked to the iterated Galois groups of unicritical polynomials $f(x) = x^d + c$ . By analyzing the Galois action on the preimages of a transcendental element $t$ , the author demonstrates that for specific choices of $K$ and $c$ , the arithmetic iterated Galois group $G_\infty(K, x^d+c, t)$ is isomorphic to the constructed group $G^P_Q$ .

Key Contributions and Results

Theorem 1 (Main Result): For a post-critically infinite unicritical polynomial $f(x) = x^d + c$ over a field $K$ (where $\text{char}(K) \nmid d$ ), the arithmetic iterated Galois group $G_\infty(K, x^d+c, t)$ is regular branch.
- Its Hausdorff dimension is explicitly calculated as $H(G) = \frac{\log(d)}{\log(d!)}$ .
- Its fixed-point proportion is given by:
  $FPP(G) = \frac{\#\{a \in I : a - 1 \in (\mathbb{Z}/d\mathbb{Z})^\times\}}{\Phi(d)}$
  where $I$ is the image of the Galois group $\text{Gal}(K(\xi)/K)$ in $(\mathbb{Z}/d\mathbb{Z})^\times$ .
- Crucial Finding: If $K = \mathbb{Q}$ and $d$ is odd, the fixed-point proportion is positive. Specifically, $FPP = \prod_{p|d} \frac{p-2}{p-1}$ . This provides the first examples outside the binary tree where a level-transitive group has both positive Hausdorff dimension and positive FPP.
Theorem 2 (General Family): The paper constructs a broader family of regular branch groups $G^P_Q$ for $d = 2^n r$ (with $r$ odd) where $Q = C_2^n \times C_r$ . These groups also exhibit positive Hausdorff dimension and, for $d \not\equiv 2 \pmod 4$ , positive fixed-point proportion.
Theorem 3 (Fixed-Point Proportion of Iterated Wreath Products): The paper provides a general algorithm to compute the FPP of any iterated wreath product $W_S$ (where $S \subseteq \text{Sym}(d)$ ). It proves that the FPP is the largest fixed point of a polynomial $f_S(x)$ of degree at most $d-1$ . It characterizes when the FPP is 0, 1, or strictly between 0 and 1 based on the transitivity and fixed-point properties of elements in $S$ .
Resolution of the Conjecture: The paper answers Jones' Question 1.1 in the negative for the class of regular branch iterated Galois groups. It demonstrates that the condition "positive Hausdorff dimension implies zero fixed-point proportion" does not hold for these groups when the degree $d$ is odd.

Significance
The paper claims to provide the first examples outside the binary rooted tree ( $d=2$ ) where a level-transitive group possesses both positive Hausdorff dimension and positive fixed-point proportion. This directly refutes the conjecture that such properties are mutually exclusive for "large" groups in the context of iterated Galois groups.

Furthermore, the work bridges the gap between abstract group theory (regular branch groups) and arithmetic dynamics (iterated Galois groups). By explicitly calculating the FPP for these arithmetic groups, the paper offers new tools for understanding the Dirichlet density of primes dividing terms in dynamical sequences, a primary application of fixed-point proportions in number theory. The construction of the family $G^P_Q$ and the derivation of the characteristic polynomial $f_S(x)$ provide a systematic framework for analyzing fixed-point proportions in groups acting on trees, extending previous results that were limited to specific cases or subgroups.

The author notes that while the binary tree case ( $d=2$ ) was previously addressed by Boston, the odd-degree case ( $d \geq 3$ ) presented a significant obstruction in previous literature, which this paper overcomes by utilizing the specific structure of unicritical polynomials and the arithmetic of cyclotomic fields.

Technical Summary: Branch Iterated Galois Groups with Positive Fixed-Point Proportion and Positive Hausdorff Dimension

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