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Santiago Radi

Publications and source records attributed to Santiago Radi.

11 recordsLinked to original sources
arXiv2026

The inverse Galois problem of iterated Galois groups and their fixed-point proportion

In 1985, Odoni initiated the study of arboreal representations and the fixed-point proportion, motivated by prime density problems in arithmetic dynamics. Since then, many questions regarding the connection between the dynamics of rational functions and the Galois groups associated to their dynamics (iterated Galois groups) have been posed. In this article, we give several contributions to this connection with the introduction of the concept of virtually mixing groups. This new concept is related to the capability of self-replication of the iterated Galois groups, and when it fails to be completely self-replicated, the failure is always by a finite index normal subgroup. It turns out that the quotient by this subgroup contains valuable information of the properties of the rational function. First, we investigate the inverse Galois problem, showing that iterated Galois groups of rational functions are always virtually mixing and the failure (the quotient by this finite-index subgroup) is related to geometric properties of the rational function. As a consequence, we prove that the rational function is induced by an endomorphism of an algebraic curve (dynamical pullback) if and only if this failure is non-trivial. Finally, we solve the main open problem of the fixed-point proportion of geometric iterated Galois groups of rational functions, by showing that the fixed-point proportion is zero when the map is not a dynamical pullback. This result has direct applications to prime density problems, proportion of periodic points in the reduction of maps and proportion of periodic points over finite fields. The proofs in this article rely on a combination of techniques and results from group theory, ergodic theory, probability, number theory, complex dynamics, energy transport in graphs, arithmetic dynamics and algebraic geometry.

math.NT↗
arXiv2026

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

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.

math.GR↗
arXiv2026

Proportion of periodic points in reduction of polynomials

In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given $K$ a number field and $f$ a rational function with coefficients in $K$, if $f_\mathfrak{p}$ denotes the reduction of $f$ modulo a prime ideal $\mathfrak{p}$ in the ring of integers of $K$, what is the limit inferior of the proportion of periodic points of $f_\mathfrak{p}$ when the norm of $\mathfrak{p}$ goes to infinity? Recent results of Fariña-Asategui and the author show that when $f$ is a polynomial of degree $d \geq 2$ non-linearly conjugate over $\mathbb{C}$ to a Chebyshev polynomial then the limit is zero. In this article, we address the remaining cases to give a complete classification of the problem in the case of polynomials.

math.NT↗
arXiv2026

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

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).

math.GR↗
arXiv2026

On the fixed-point proportion of self-similar groups

We prove that super strongly fractal groups acting on regular rooted trees have null fixed-point proportion. In particular, we show that the fixed-point proportion of an infinite family of iterated monodromy groups of exceptional complex polynomials have the same property. The proof uses the approach of Rafe Jones in [15] based on martingales and a recent result of the first author on the dynamics of self-similar groups [6].

math.GR↗
arXiv2026

A just-infinite iterated monodromy group without the congruence subgroup property

We prove that the iterated monodromy group of the polynomial $z^2+i$ is just-infinite, regular branch and does not have the congruence subgroup property. This yields the first example of an iterated monodromy group of a polynomial with these properties. Additional information is provided about the congruence kernel, rigid kernel and branch kernel of this group.

math.GR↗
arXiv2026

Fixed-point proportion of geometric iterated Galois groups

In 1980, Odoni initiated the study of the fixed-point proportion of iterated Galois groups of polynomials motivated by prime density problems in arithmetic dynamics. The main goal of the present paper is to completely settle the longstanding open problem of computing the fixed-point proportion of geometric iterated Galois groups of polynomials. Indeed, we confirm the well-known conjecture that Chebyshev polynomials are the only complex polynomials whose geometric iterated Galois groups have positive fixed-point proportion. Our proof relies on methods from group theory, ergodic theory, martingale theory and complex dynamics. This result has direct applications to the proportion of periodic points of polynomials over finite fields. The general framework developed in this paper applies more generally to rational functions over arbitrary fields and generalizes, via a unified approach, previous partial results, which have all been proved with very different methods.

math.NT↗
arXiv2025

Groups of finite type: classification and structural properties

Groups of finite type (also called finitely constrained groups), introduced by Grigorchuk, are known to be the closure of regular branch groups. This article explores many of their properties. Firstly, we prove that being finitely generated, just-infinite and strongly complete are equivalent in a vast family of groups of finite type. As a consequence, we prove that the closure of the Hanoi towers group on 3 pegs is just-infinite although the group itself is not. Secondly, we improve the algorithm given by Bondarenko and Samoilovych in [9], to compute all the groups of finite type of a given depth and acting on a given tree. We use this to find the groups of finite type acting on the ternary tree with depth 2 and 3. Thirdly, we give a sufficient condition for a group generated by a finite automaton of Mealy type to have as closure a group of finite type. This allows us to identify groups of finite type as the closure of explicit groups generated by a finite automaton. Lastly, we give an algorithm to prove whether two groups of finite type are isomorphic. With this result, we classify groups of finite type up to isomorphism in the binary tree for depths 2, 3 and 4 and in the ternary tree for depths 2 and 3.

math.GR↗
arXiv2025

Random subgroups of branch groups

We show that independent Haar-random elements in a super strongly fractal branch profinite group generate a free subgroup acting freely on the boundary of the tree. This improves a previous result of Abért (2005) for weakly branch profinite groups, where independent random elements were shown to generate free subgroups acting only almost freely on the boundary. Our result also generalizes the analogous result of Abért and Virág (2005) for iterated wreath products.

math.GR↗