arXiv subjects

Emanuele Rodaro

Publications and source records attributed to Emanuele Rodaro.

At least 19 recordsLinked to original sources

arXiv2026

A graph-theoretical characterisation of subgroups of Thompson's group $V$

We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$: a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in $V$, providing evidence towards Lehnert's conjecture. - Each finitely generated subgroup of $V$ is either virtually abelian, or contains a free non-abelian semigroup. It follows that groups of intermediate growth do not embed in Thompson's $V$. We further study the relation between transition groups defined by graphs that are limits or covers of each others, and prove properties of transition groups of context-free graphs of polynomial growth. Finally, we prove that the Basilica and Hano\"i Towers groups do not embed in $V$. This uses the geometry of Schreier graphs of the natural actions of these groups and of Thompson's $V$.

math.GR↗

arXiv2026

On totally synchronizing graphs

A coloring of a finite $k$-out directed graph $G$ is viewed as a deterministic complete automaton with state set $V(G)$. The graph $G$ is called \emph{totally synchronizing} if every coloring is synchronizing. We prove that total synchronization imposes strong restrictions on symmetry: if $G$ is strongly connected and totally synchronizing, then $Aut(G)$ contains no semiregular element; in particular, if $|Aut(G)|$ is divisible by a prime $p>k$, then $G$ is not totally synchronizing. We then give general constructions of strongly connected $k$-out graphs with prescribed quotients and prescribed automorphism group that are \emph{not} totally synchronizing. On the quotient side, we relate graph congruences to strong lumpability of the uniform random walk on $G$ and introduce \emph{totally simple} graphs, characterized by the absence of nontrivial congruences. In this setting we obtain a Perron--Frobenius sufficient condition for total synchronization: a strongly connected non-lumpable graph whose integer Perron--Frobenius eigenvector admits at most one nontrivial equipartition is totally synchronizing. Finally, we show that deciding whether a primitive $k$-out graph admits a non-synchronizing coloring is NP-complete, resolving an open problem of Gusev--Szyku{\l}a, and prove NP-completeness of deciding whether a graph admits a nontrivial Eulerian lumping.

math.CO↗

arXiv2026

Topological indices on self-similar graphs generated by groups

In this paper, we determine precise formulas for the diameters, the number of perfect matchings, and the Tutte polynomials for an infinite family of finite graphs, namely the Schreier graphs of tree automaton groups, also called tree graph automata. This enables us to easily find the number of spanning trees, spanning forests, and an explicit form for the chromatic polynomials. In the second part of the paper, we provide the precise values for the Wiener and Szeged index of any tree graph automaton.

math.CO↗

arXiv2025

The hereditariness problem for the \v{C}ern\'y conjecture

This paper addresses the lifting problem for the \v{C}ern\'y conjecture: namely, whether the validity of the conjecture for a quotient automaton can always be transferred (or "lifted") to the original automaton. Although a complete solution remains open, we show that it is sufficient to verify the \v{C}ern\'y conjecture for three specific subclasses of reset automata: radical, simple, and quasi-simple. Our approach relies on establishing a Galois connection between the lattices of congruences and ideals of the transition monoid. This connection not only serves as the main tool in our proofs but also provides a systematic method for computing the radical ideal and for deriving structural insights about these classes. In particular, we show that for every simple or quasi-simple automaton $\mathcal{A}$, the transition monoid $\text{M}(\mathcal{A})$ possesses a unique ideal covering the minimal ideal of constant (reset) maps; a result of similar flavor holds for the class of radical automata.

cs.FL↗

arXiv2025

Horofunctions of infinite Sierpinski polygon graphs

Generalizing works of D'Angeli and Donno, we describe, starting from an infinite sequence over $r$ letters with $r \neq 4i$ and $i \in \mathbb{N}$, a sequence of pointed finite graphs. We study the pointed Gromov-Hausdorff limit graphs giving a description of isomorphim classes in terms of dihedral groups and providing insights on the horofunction boundaries in terms of Busemann and non-Busemann points.

math.CO↗

arXiv2025

Freeness Obstructions for Self-Similar Groups Acting on Rooted Trees

We prove two general obstructions to freeness for self-similar groups and derive consequences for automaton groups. First, let $\mathcal A$ be a finite invertible Mealy automaton with a co-accessible identity state. Then every finitely generated subgroup of $G(\mathcal A)$ that is either self-similar and transitive on the first level, or level-transitive on the rooted tree, is cyclic or non-free. This answers a question of Grigorchuk and partially extends Sidki's obstruction for automata of polynomial activity. The methods also yield effective, quantitative, and geometric consequences for relations and level Schreier graphs. Second, we study the relation between freeness and bireversibility. We prove that if a finite reduced reversible invertible Mealy automaton generates a non-abelian free group, then its dual has a bireversible connected component. If the states form a free basis and the action is transitive on the first level, then the automaton itself is bireversible. Finally, over a binary alphabet, the free-basis assumption can also be removed: every finite reduced invertible Mealy automaton generating a non-abelian free group and acting level-transitively is bireversible.

math.GR↗

arXiv2024

On the ET0L subgroup membership problem in bounded automata groups

We are interested in the subgroup membership problem in groups acting on rooted $d$-regular trees and a natural class of subgroups, the stabilisers of infinite rays emanating from the root. These rays, which can also be viewed as infinite words in the alphabet with d letters, form the boundary of the tree. Stabilisers of infinite rays are not finitely generated in general, but if the ray is computable, the membership problem is well posed and solvable. The main result of the paper is that, for bounded automata groups, the membership problem in the stabiliser of any ray that is eventually periodic as an infinite word, forms an ET0L language that is constructable. The result is optimal in the sense that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be context-free. As an application, we give a recursive formula for the associated generating function, aka the Green function, on the corresponding infinite Schreier graph.

math.GR↗

arXiv2024

Context-free graphs and their transition groups

Starting from context-free inverse graphs, we introduce a new class of groups and study their structural properties. We establish closure properties, show that their co-word problems are context-free, analyze torsion elements, and realize them as subgroups of the asynchronous rational group. Context-freeness is preserved under a generalized free product of graphs, and using this construction we provide examples of groups that are not residually finite or not poly-context-free, making them relevant for testing the Lehnert and Brough conjectures. Moreover, we investigate how small local modifications of a graph affect the global structure of the transition group, showing that for locally quasi-transitive graphs with infinite orbits, the transition group decomposes into a highly structured quotient by a bounded torsion subgroup, showing strong global constraints induced by local graph properties.

math.GR↗

arXiv2024

The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity

We extend the notion of activity for automaton semigroups and monoids introduced by Bartholdi, Godin, Klimann and Picantin to a more general setting. Their activity notion was already a generalization of Sidki's activity hierarchy for automaton groups. We show that the language of $\omega$-words with infinite orbits is effectively a deterministic B\"uchi language for automata with bounded extended activity, which yields decidability of the finiteness problem for complete automaton semigroups and monoids of bounded activity (solving an open problem by Bartholdi, Godin, Klimann and Picantin). In fact, we obtain a stronger result also covering finitely generated subsemigroups.

cs.FL↗

arXiv2024

The Freeness Problem for Automaton Semigroups

We show that the freeness problems for automaton semigroups and for automaton monoids are undecidable and, thereby, solve an open problem listed by Grigorchuk, Nekrashevych and Sush\-chansk\u{\i}i. We achieve this using a new technique to encode Post's Correspondence Problem into automaton semigroups and monoids and our result even holds if we restrict the alphabet of the input automata to a constant size. The encoding allows us to precisely control the relations in the generated semigroup/monoid and the construction is quite versatile. In fact, we obtain further undecidability results on various semigroup notions (left cancellativity, equidivisibility and extending homomorphisms). Our construction can also be adapted to show that the free presentation problem for automaton monoids is undecidable (and yields a weaker statement in the semigroup case).

cs.FL↗

arXiv2023

Generalizations of the Muller-Schupp theorem and tree-like inverse graphs

We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph $Γ$, being quasi-isometric to a tree, or context-free (finitely many end-cones types), or having the automorphism group $Aut(Γ)$ that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky-Schützenberger that is fundamental in solving a weaker version of a conjecture of T. Brough which also extends Muller and Schupp' result to the class of groups that are virtually finitely generated subgroups of direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.

math.GR↗

arXiv2023

Applications of Automaton Groups in Cryptography

In 1991 the first public key protocol involving automaton groups has been proposed. In this paper we give a survey about algorithmic problems around automaton groups which may have potential applications in cryptography. We then present a new public key protocol based on the conjugacy search problem in some families of automaton groups. At the end we offer open problems that could be of interest of group theorists and computer scientists in this direction.

math.GR↗

arXiv2022

The Self-Similarity of Free Semigroups and Groups

We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area.

math.GR↗

arXiv2022

On a class of poly-context-free groups generated by automata

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is isomorphic to the partially commutative monoid associated with the complement of the line graph of the defining tree. After that, we generalize these groups by introducing the quite broad class of reducible automaton groups, which lies in the class of contracting automaton groups without singular points. We give a general structure theorem that shows that all reducible automaton groups are direct limit of poly-context-free groups which are virtually subgroups of the direct product of free groups; notice that this result partially supports a conjecture by T. Brough. Moreover, we prove that tree automaton groups with at least two generators are not finitely presented and they are amenable groups, which are direct limit of non-amenable groups.

math.GR↗

arXiv2021

On an uncountable family of graphs whose spectrum is a Cantor set

For each $p\geq 1$, the star automaton group $\mathcal{G}_{S_p}$ is an automaton group which can be defined starting from a star graph on $p+1$ vertices. We study Schreier graphs associated with the action of the group $\mathcal{G}_{S_p}$ on the regular rooted tree $T_{p+1}$ of degree $p+1$ and on its boundary $\partial T_{p+1}$. With the transitive action on the $n$-th level of $T_{p+1}$ is associated a finite Schreier graph $Γ^p_n$, whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as limits of the sequence $\{Γ_n^p\}_{n\geq 1}$ in the Gromov-Hausdorff topology. We obtain an explicit description of the spectrum of the graphs $\{Γ_n^p\}_{n\geq 1}$. Then, by using amenability of $\mathcal{G}_{S_p}$, we prove that the spectrum of each infinite Schreier graph is the union of a Cantor set of zero Lebesgue measure, which is the Julia set of the quadratic map $f_p(z) = z^2-2(p-1)z -2p$, and a countable collection of isolated points supporting the KNS spectral measure. We also give a complete classification of the infinite Schreier graphs up to isomorphism of unrooted graphs, showing that they may have $1$, $2$ or $2p$ ends, and that the case of $1$ end is generic with respect to the uniform measure on $\partial T_{p+1}$.

math.GR↗

arXiv2020

Infinite Automaton Semigroups and Groups Have Infinite Orbits

We show that an automaton group or semigroup is infinite if and only if it admits an $ω$-word (i. e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups are infinite if and only if there is an $ω$ -word with an infinite orbit under their action. The proof also shows some interesting connections between the automaton semigroup and its dual. Finally, our result is interesting from an algorithmic perspective as it allows for a reformulation of the finiteness problem for automaton groups and semigroups.

cs.FL↗

arXiv2020

Graph automaton groups

In this paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two elements and is amenable of exponential growth. We also highlight a connection between our construction and the right-angled Artin groups. We then study the Schreier graphs associated with the self-similar action of these automaton groups on the regular rooted tree. We explicitly determine their diameter and their automorphism group in the case where the initial graph is a path. Moreover, we show that the case of cycles gives rise to Schreier graphs whose automorphism group is isomorphic to the dihedral group. It is remarkable that our construction recovers some classical examples of automaton groups like the Adding machine and the Tangled odometer.

math.GR↗

arXiv2020

On the Orbits of Automaton Semigroups and Groups

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata. Then, we look at $ω$-word (i.e. right infinite words) with a finite orbit. We show that every automaton yielding an $ω$-word with a finite orbit already yields an ultimately periodic one, which is not periodic in general, however. On the algorithmic side, we observe that it is not possible to decide whether a given periodic $ω$-word has an infinite orbit and that we cannot check whether a given reversible and complete automaton admits an $ω$-word with a finite orbit, a reciprocal problem to the finiteness problem for automaton semigroups in the reversible case. Finally, we look at automaton groups generated by reversible but not bi-reversible automata and show that many words have infinite orbits under the action of such automata.

cs.FL↗