Searcharxiv⌕ Search

arXiv subjects

Jorge Fariña-Asategui

Publications and source records attributed to Jorge Fariña-Asategui.

At least 19 recordsLinked to original sources
arXiv2026

Uncountably many local isomorphism types of compactly generated simple groups

A major open question in the theory of locally compact groups is the following. Let $\mathscr{S}$ be the class of non-discrete compactly generated totally disconnected locally compact groups that are topologically simple. Is the number of local isomorphism classes of groups in $\mathscr{S}$ uncountable? We have answered this question, showing that there are $2^{\aleph_0}$ local isomorphism classes in $\mathscr{S}$. Our result was obtained without the use of artificial intelligence; it arose from a problem session that ran over several days at the workshop "Branch groups: subgroups, rigidity, topologies" at the Universidad Complutense de Madrid, organised by Dominik Francoeur, Alejandra Garrido and Tatiana Nagnibeda.

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

Arboreal Galois representations of rational functions: fixed-point proportion and the extension problem

We give an explicit description of the arithmetic-geometric extension of iterated Galois groups of rational functions. This yields a complete solution to the extension problem when either the arithmetic or the geometric iterated Galois group is branch, answering a question of Adams and Hyde. Furthermore, we obtain a sufficient condition for the arithmetic iterated Galois group of a rational function to have positive fixed-point proportion, which further applies in many instances to the specialization to non strictly post-critical points. In particular, this holds for all unicritical polynomials of odd degree, which greatly generalizes a result of Radi for the polynomial $z^d+1$. Lastly, we obtain the first family of groups acting on the $d$-adic tree whose fixed-point process becomes eventually $d$ for any $d\ge 2$ with positive probability. What is more, these groups are fractal and branch and thus positive-dimensional; hence they yield the first family of counterexamples to a conjecture of Jones for every $d$-adic tree.

math.NT↗
arXiv2026

Virtual retracts in groups acting on rooted trees

We study virtual retracts in groups acting on rooted trees. We show that finitely generated branch groups do not have the local retraction (LR) property. Furthermore, we specialize to iterated monodromy groups of post-critically finite quadratic complex polynomials and show that the (LR) property characterizes, among post-critically finite quadratic complex polynomials, those with a euclidean orbifold, i.e. the powering map and the Chebyshev polynomial. Lastly, we show that periodic quadratic complex polynomials provide new examples of pro-$2$ groups with complete finitely generated Hausdorff spectrum.

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

Non-closed subgroups of weakly branch groups

For a weakly branch group $G$ acting on a regular enough rooted tree, we provide two constructions of continuous families of distinct subgroups that are not closed in the profinite topology on $G$. On the one hand, we construct a continuous family of distinct non-closed subgroups such that each $H$ in the family is not ERF, that is, contains subgroups not closed in the profinite topology on $H$. On the other hand, under an additional assumption on $G$, we construct a continuous family of ERF subgroups which are not closed in the congruence (and in the profinite) topology on $G$.

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↗
arXiv2025

Weakly branch actions: first-order theory, rigidity and Boston's conjecture

We disprove a well-known conjecture of Boston (2000), which claims that a just-infinite pro-$p$ group is branch if and only if it admits a positive-dimensional embedding in the group of $p$-adic automorphisms. This is obtained as a result of a comprehensive study of the rigidity of branch actions. Firstly, we generalize the notion of the structure graph, introduced by Wilson in 2000, to weakly branch groups and use it to prove several results on the first-order theory of weakly branch groups, extending previous results of Wilson on branch groups. Secondly, we completely characterize the rigidity of weakly branch and branch actions on arbitrary spherically homogeneous rooted trees, extending previous partial results (for branch actions) by Hardy, Garrido, Grigorchuk and Wilson. Moreover, we prove that rigidity of a weakly branch group is equivalent to rigidity of its closure in the full automorphism group. Thirdly, we extend greatly the sufficient conditions $(*)$ and $(**)$ of Grigorchuk and Wilson, which leads to a complete and very easy-to-check characterization of the rigidity of the weakly branch actions of a fractal group of $p$-adic automorphisms. We further establish the first connection in the literature between the Hausdorff dimension of a weakly branch action and its rigidity. Lastly, we put everything together to show that the zero-dimensional just-infinite branch pro-$p$ groups introduced recently by the author admit rigid branch actions on a tree obtained by deletion of levels. This, together with previous results of the author, shows that these groups are indeed counterexamples to the aforementioned conjecture of Boston.

math.GR↗
arXiv2025

Arboreal representations of linear groups

Abért and Virág conjectured in 2005 that any embedding of a linear group over a pro-$p$ domain into the group of $p$-adic automorphisms $W_p$ should be zero-dimensional. We prove their conjecture in greater generality, namely for embeddings of linear groups over any integral domain into the automorphism group of a bounded rooted tree.

math.GR↗
arXiv2025

On a question of Abért and Virág

Abért and Virág proved in 2005 that the Hausdorff dimension of a non-trivial normal subgroup of a level-transitive 1-dimensional subgroup of the group of $p$-adic automorphisms $W_p$ is always 1. They further asked whether the same holds replacing 1-dimensional with positive dimensional. On the one hand, we provide a negative answer in general by giving counterexamples where the non-trivial normal subgroups are not all 1-dimensional. Furthermore, these counterexamples are pro-$p$ subgroups of $W_p$ with positive Hausdorff dimension in $W_p$ but with non-trivial center, and thus not weakly branch. On the other hand, we restrict ourselves to the class of self-similar groups and answer the question of Abért and Virág in the positive in this case. Along the way, we generalize a result of Abért and Virág on the closed subgroups of $W_p$ being perfect in the sense of Hausdorff dimension to closed subgroups of any iterated wreath product $W_H$ and show that self-similar positive-dimensional subgroups of $W_H$ do not satisfy any group law.

math.GR↗
arXiv2025

Markov processes associated to fractal branch groups

The author introduced recently a new natural construction which associates a measure-preserving dynamical system to any fractal profinite group. Here, we investigate these measure-preserving dynamical systems under the extra assumption on the groups to be branch. First, we compute their $f$-invariant, a measure-conjugacy invariant introduced by Bowen, and show that they are Markov processes over free semigroups in the sense of Bowen. Secondly, we show that fractal branch profinite groups with the same Hausdorff dimension and whose associated measure-preserving dynamical systems have the same $f$-invariant yield isomorphic Markov processes.

math.DS↗
arXiv2025

Restricted Hausdorff spectra of $q$-adic automorphisms

Firstly, we completely determine the self-similar Hausdorff spectrum of the group of $q$-adic automorphisms where $q$ is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products. Secondly, we provide for every prime $p$ the first examples of just infinite branch pro-$p$ groups with zero Hausdorff dimension in $Γ_p$, giving strong evidence against a well-known conjecture of Boston.

math.GR↗
arXiv2025

Cyclicity, hypercyclicity and randomness in self-similar groups

We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms in contracting discrete automata groups. In the profinite setting we prove that fractal profinite groups may be regarded as measure-preserving dynamical systems and derive a sufficient condition for the ergodicity and the mixing properties of these dynamical systems. Furthermore, we show that a Haar-random element in a super strongly fractal profinite group is hypercyclic almost surely as an application of Birkhoff's ergodic theorem for free semigroup actions.

math.GR↗
arXiv2025

Branch actions and the structure lattice

J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group $G$ on a spherically homogeneous rooted tree $T$ there is a canonical $G$-equivariant isomorphism between the Boolean algebra associated with the structure lattice of $G$ and the Boolean algebra of clopen subsets of the boundary of $T$.

math.GR↗
arXiv2025

On finitely generated Hausdorff spectra of groups acting on rooted trees

We answer two longstanding questions of Klopsch (1999) and Shalev (2000) by proving that the finitely generated Hausdorff spectrum of the closure of a finitely generated regular branch group with respect to the level-stabilizer filtration is the full interval [0,1]. Previous work related to the above questions include the celebrated result of Abért and Virág on the completeness of the finitely generated Hausdorff spectrum of the group of $p$-adic automorphisms $W_p$. We extend their result to any level-transitive iterated wreath product acting on a regular rooted tree, and more importantly, provide the first explicit examples of finitely generated subgroups with prescribed Hausdorff dimension. In fact, in $W_p$ such subgroups can be taken to be 2-generated, giving further evidence to a conjecture of Abért and Virág. Furthermore, we extend a well-known result of Klopsch (1999) for branch groups to closed level-transitive groups with fully dimensional rigid stabilizers: these have full Hausdorff spectrum with respect to the level-stabilizer filtration. In turn, we determine, for the first time, the Hausdorff spectum of many well-studied families of weakly branch groups. As an additional new contribution, we define {\it nice filtrations} (which include the level-stabilizer filtration). In fact, the first and the last result can be proven more generally with respect to these filtrations (under possibly some further assumptions). Finally, we also consider two standard filtrations, namely the $2$-central lower series, and the dimension subgroup or Jenning-Zassenhaus series in the closure of the first Grigorchuck group, and show that its finitely generated Hausdorff spectra with respect to these filtrations is the full interval [0,1], answering a recent question of de las Heras and Thillaisundaram (2022). This is obtained via a reduction to properties concerning to nice filtrations.

math.GR↗