I study geometric and combinatorial group theory.
I am currently interested in:
Image of the fractal at the end of the animation by J. Belk
My papers
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 or 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.
We study a family of Thompson-like groups built as rearrangement groups of fractals from [BF19], each acting on a Wazewski dendrite. Each of these is a finitely generated group that is dense in the full group of homeomorphisms of the dendrite (studied in [DM19]) and has infinite-index finitely generated simple commutator subgroup, with a single possible exception.
More properties are discussed, including finite subgroups, the conjugacy problem, invariable generation and existence of free subgroups. We discuss many possible generalizations, among which we find the Airplane rearrangement group TA.
Despite close connections with Thompson's group F, dendrite rearrangement groups seem to share many features with Thompson's group V.
We describe a method for solving the conjugacy problem in a vast class of rearrangement groups of fractals, a family of Thompson-like groups introduced in 2019 by Belk and Forrest. We generalize the methods of Belk and Matucci for the solution of the conjugacy problem in Thompson groups F, T and V via strand diagrams.
In particular, we solve the conjugacy problem for the Basilica, the Airplane, the Vicsek and the Bubble Bath rearrangement groups and for the groups QV, Q̃V, QT, Q̃T and QF, and we provide a new solution to the conjugacy problem for the Houghton groups and for the Higman-Thompson groups, where conjugacy was already known to be solvable.
Our methods involve two distinct rewriting systems, one of which is an instance of a graph rewriting system, whose confluence in general is of interest in computer science.
A Class of Rearrangement Groups that are not Invariably Generated
(Bull. London Math. Soc. 56.6, pp. 2115–2131)
joint work with Davide Perego
arXiv link
A group G is invariably generated if there exists a subset S ⊆ G such that, for every choice gs ∈ G for s ∈ S, the group G is generated by { sgs ∣ s ∈ 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.
Generation and Simplicity in the Airplane Rearrangement Group
(Groups Geom. Dyn. 18.2, pp. 603–634)
adapted from my master thesis
arXiv link
We study the group TA of rearrangements of the Airplane limit space introduced by Belk and Forrest in [3]. We prove that TA is generated by a copy of Thompson's group F and a copy of Thompson's group T, hence it is finitely generated. Then we study the commutator subgroup [TA, TA], proving that the abelianization of TA is isomorphic to Z and that [TA, TA] is simple, finitely generated and acts 2-transitively on the so-called components of the Airplane limit space. Moreover, we show that TA is contained in T and contains a natural copy of the Basilica rearrangement group TB studied in [2].
Talks I gave
Group Theory in Florence IV - 1st-3th July 2024
Insalata di Matematica, invited talk - 9th May 2024
SPIEDini, invited talk - 14th March 2024
Groups and Actions seminar (Université Paris-Saclay), invited talk - 12th February 2024
GAGTA 2024 - 5th-9th February 2024
Congresso UMI, Algebra section - 4th-9th September 2023
Teoria dei Gruppi a Paestum - 6-7th July 2023
Groups and Topological Groups in Milano - 15th-16th June 2023
Manifolds and groups in Bologna workshop, invited talk - 22nd-23rd March 2023
Geometry and Topology Seminar of the University of Glasgow, invited talk - 6th February 2023
Group Theory seminar of ENS, Paris (recording), invited talk - 19th October 2022
BiLux 2022 - 3rd-4th Ocotber 2022
Events I organized
GABY 2024 (biennial algebra workshop at Milano-Bicocca) - June 2024
AlBicocca (Algebra seminars of Milano-Bicocca) - 2023-ongoing
Insalate di Matematica (seminars at Milano-Bicocca organized by PhD students) - 2022-2024
GABY 2022 (biennial algebra workshop at Milano-Bicocca) - June 2022
Other conferences I attended
Young Geometric Group Theory XII at University of Bristol - April 2024
Discrete Mathematics and Computer Science at CIRM - January / February 2024
Groups of Thompson and their relatives at University of Magdeburg - September 2023
North British Geometric Group Theory Seminar at University of Glasgow - April 2023
Young Geometric Group Theory XI at University of Münster - February 2023
Discrete and locally compact groups: geometry and analysis at ENS, Lyon - November 2022
Marc Burger's Prime at ETH, Zurich - May / June 2022
Self-similarity of groups, trees and fractals at IHP, Paris - June / July 2022
Graduate School on Geometric Group Theory and Low Dimensional Topology at ICMAT, Madrid - May 2022
Young Geometric Group Theory X (online) - July 2021
GAGTA 2021 (online) - June 2021
Teaching I did
Teaching support for Logica e Algebra at Politecnico di Milano, Scuola di Ingegneria Industriale e dell'Informazione - AY 2023/24
Tutoring for Istituzioni di Matematica 1 at Milano-Bicocca, Scuola d Ottica e Optometria - AY 2023/24
Miscellanea
Journal referee for Results in Nonlinear Analysis