arXiv subjects

Davide Perego

Publications and source records attributed to Davide Perego.

13 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

Period growth and co-context-free groups

We study period growth in co-context-free groups, giving general results and looking at specific examples such as Thompson groups $T$ and $V$ and the Houghton groups $H_m$. Along the way, we give a refined upper bound on the word metric in Thompson $V$, as well as efficient algorithms to determine if elements of $V$ are torsion, and compute their order. We also adapt our algorithm to compute the rotation number of elements of $T$ and answer a question of D. Calegari.

math.GR↗

arXiv2025

Canonical Reduction Systems in Artin-Tits groups of spherical type

We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period.

math.GR↗

arXiv2025

Limit spaces of vertex and edge replacement systems

We introduce and study VERSs (vertex and edge replacement systems) as a technology of graph expansions. We consider its history graph, an augmented tree that records each graph expansion, and we provide sufficient conditions under which it is hyperbolic. When hyperbolic, its Gromov boundary is what we call the limit space of the VERS. We provide three examples from different areas of mathematics: Schreier graphs and limit spaces of finitely generated contracting self-similar groups, injective post-critically finite iterated function systems and limit spaces of edge replacement systems.

math.CO↗

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

Orderable Thompson-like groups arising from Ore categories

We give sufficient conditions for left- and bi-orderability of fundamental groups of Ore categories in terms of indirect factors, including Thompson groups and many of their generalizations. Besides recovering known results, we prove that braided groups of fractions of digit rewriting systems (which generalize braided Thompson groups to the wider setting of topological full groups of edge shift) are left-orderable, and that their purely braided counterparts are bi-orderable. In particular, the braided Houghton groups are left-orderable.

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

Eventually Self-Similar Groups acting on Fractals

Generalizing work by Belk and Forrest, we develop almost expanding hyperedge replacement systems that build fractal topological spaces as quotients of edge shifts under certain ``gluing'' equivalent relations. We define ESS groups, which are groups of homeomorphisms of these spaces that act as a finitary asynchronous transformations followed by self-similar ones, akin to the action of Scott-R\"{o}ver-Nekrashevych groups on the Cantor space. We provide sufficient conditions for finiteness properties of such groups, which allow us to show that the airplane and dendrite rearrangement groups have type $F_\infty$, that a group combining dendrite rearrangements and the Grigorchuk group is finitely generated, and that certain ESS groups of edge shifts have type $F_\infty$ (partially addressing a question of Deaconu), in addition to providing new proofs of previously known results about several Thompson-like groups.

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

Rational Gluing in Edge Replacement Systems

In this paper, we prove the rationality of the gluing relation of edge replacement systems, which were introduced for studying rearrangement groups of fractals. More precisely, we describe an algorithmic procedure for building a finite state automaton that recognizes pairs of equivalent sequences that are glued in the fractal. This fits in recent interest towards the rationality of gluing relations on totally disconnected compact metrizable spaces.

math.GR↗

arXiv2023

Rationality of the Gromov Boundary of Hyperbolic Groups

In [BBM21], Belk, Bleak and Matucci proved that hyperbolic groups can be seen as subgroups of the rational group. In order to do so, they associated a tree of atoms to each hyperbolic group. Not so many connections between this tree and the literature on hyperbolic groups were known. In this paper, we prove an atom-version of the fellow traveler property and exponential divergence, together with other similar results. These leads to several consequences: a bound from above of the topological dimension of the Gromov boundary, the definition of an augmented tree which is quasi-isometric to the Cayley graph and a synchronous recognizer which described the equivalence relation given by the quotient map defined from the end of the tree onto the Gromov boundary.

math.GR↗

arXiv2022

A Class of Rearrangement Groups that are not Invariably Generated

A group $G$ is invariably generated if there exists a subset $S \subseteq G$ such that, for every choice $g_s \in G$ for $s \in S$, the group $G$ is generated by $\{ s^{g_s} \mid s \in S \}$. In [GGJ16] Gelander, Golan and Juschenko showed that Thompson groups $T$ and $V$ are not invariably generated. Here we generalize this result to the larger setting of rearrangement groups, proving that any subgroup of a rearrangement group that has a certain transitive property is not invariably generated.

math.GR↗

arXiv2002

Limits on $\nu_{\tau}$ mass by DELPHI

A limit on the tau neutrino mass is obtained using all the $Z^{0} \to \tau^{+} \tau^{-}$ data collected at LEP by the DELPHI detector between 1992 and 1995. In this analysis events in which one of the taus decays into one charged particle, while the second $\tau$ decays into f{}ive charged pions (1-5 topology) have been used. The neutrino mass is determined from a bidimensional \fit ~on the invariant mass $m^{*}_{5 \pi}$ and on the energy $E_{5 \pi}$ of the f{}ive $\pi^{\pm}$ system. The result found is $m_{\nu_{\tau}}<$ 48.0 MeV/$c^{2}$ at 95% conf{}idence level.

hep-ex↗