# Algebra Seminar

In the academic year 2023/2024, as in the previous one, l'Al@Bicocca will continue to be both on site in Bicocca and online!

The Algebra Seminar will take place in the Department of Mathematics and Applications of Milano-Bicocca, usually in room 3014, and will be streamed online using WebEx.

To join a meeting, please send an email to albicocca (AT) unimib (DOT) it at least 4 hours before the starting time of the talk you wish to attend.

The time indicated in this webpage is the local time in Milan, which corresponds to UTC +1.

## Next Talk

17 April 2024, 11.00-12.00 am (UTC+1) - Islam Foniqi (University of East Anglia)

Title: Decision problems in one-relator monoids

## Organizers

Marco Barbieri, Alberto Cassella, Giulia Dal Verme, Nicola Grittini, Francesco Matucci, Matteo Tarocchi

Past organizers: Valentina Grazian, Francesco Matucci

## Past Talks

## A.Y. 2023/24

3 April 2024 - Joy Morris (University of Lethbridge)

Title: Recent results on the Cayley Isomorphism property

Abstract and poster at this link.

Recording available at this link.

20 March 2024 - Nicola Grittini (University of Milano-Bicocca)

Title: Character degrees and character values

Abstract and poster at this link.

Recording available at this link.

7 March 2024 - Pavel Zalesski (Universidade de Brasilia) - AlBicocca flash seminar!

Title: Detecting the geometry of 3-manifolds by finite quotients of its fundamental group

Abstract and poster at this link.

Recording available at this link.

6 March 2024 - Daniele Nemmi (University of Padova)

Title: Graphs encoding properties of finite groups

Abstract and poster at this link.

Recording available at this link.

22 February 2024 - Giulia Dal Verme (University of Milano-Bicocca)

Title: A groupoid C*-algebraic Bass-Serre theorem

Abstract and poster at this link.

Recording available at this link.

1 February 2024 - Alberto Cassella (University of Milano-Bicocca and University of Zaragoza)

Title: Hypercubical groups

Abstract and poster at this link.

Recording available at this link.

18 January 2024 - Andoni Zozaya (University of Ljubljana)

Title: Linearity of profinite analytic groups

Abstract and poster at this link.

Recording available at this link.

21 December 2023 - Daniele Dona (Alfréd Rényi Institute of Mathematics, Budapest)

Title: Diameter of classical groups over finite fields

Abstract and poster at this link.

Recording available at this link.

30 November 2023 - Lander Guerrero Sánchez (Universidad del País Vasco / Euskal Herriko Unibertsitatea)

Titolo: Computing differentials in spectral sequences

Abstract and poster at this link.

Recording available at this link.

16 November 2023 - Luca Mariot (University of Twente)

Title: Counting Coprime Polynomials over Finite Fields with Formal Languages and Compositions of Natural Numbers

Abstract and poster at this link.

Recording available at this link.

9 November 2023 - Marina Avitabile (Università degli Studi di Milano-Bicocca)

Title: On Some Coefficients of the Artin-Hasse series modulo a prime

Abstract and poster at this link.

Recording available upon request.

## A.Y. 2022/23

26 July 2023 AlBicocca Flash Workshop

2.30pm - Collin Bleak (University of St Andrews)

Title: 3/2’s generation for Thompson-esque groups

Abstract: We borrow from the theory of finite simple groups, and ask: Does every non-trivial element f of a simple group G admit a co-generator? That is, an element g s.t. G = < f, g >? In the case of R. Thompson's groups T and V, the answer is "Yes". Indeed, for G in {T,V} we have the stronger result that there is an element z so that for any non-trivial element f in G we can find an element c in G with G = <f,z^c>. We will also discuss some developing work generalising these results to the wide class of finitely generated vigorous simple groups. Joint with Casey Donovan, Scott Harper, James Hyde, and Rachel Skipper.

3.30pm - James Belk (University of Glasgow)

Title: Finite Germ Extensions

Abstract: Sometimes a homeomorphism has finitely many points at which it exhibits irregular or "singular" behavior. For example, a homeomorphism of an interval might consist of infinitely many linear pieces that accumulate at finitely many singular points. Elements of Grigorchuk's group can also be regarded as having finitely many singular points when acting on the Cantor set. In this talk, we describe a general theorem on finiteness properties for Thompson-like groups of homeomorphisms that have finitely many singular points. Among other applications, we give the first example of a finitely presented group that contains every countable abelian group. This is joint work with James Hyde and Francesco Matucci.

Poster: Al@Bicocca-Flash.pdf

07/06/2023, 2.45pm - Luca Sabatini (Alfréd Rényi Institute of Mathematics - Budapest)

Title: On groups with large verbal quotients

Abstract: Let w = w(x1, ..., xn) be a word, i.e. an element of the free group F = <x1, ..., xn>. The verbal subgroup w(G) of a group G is the subgroup generated by the set {w(x1, ..., xn) : x1, ..., xn ∈ G} of all w-values in G. Following J. González-Sánchez and B. Klopsch, a group G is w-maximal if |H : w(H)| < |G : w(G)| for every H < G. These objects are useful in particular for the following reason: if a finite group contains a large section H/N such that w(H/N) is trivial, then it contains a large w-maximal subgroup. In this talk I will give new results on w-maximal groups, and study the weaker condition in which the inequality in the definition is not strict.

Poster: Al@Bicocca - Sabatini.pdf

24/05/2023, 3.00pm - Alexander Bishop (Université de Geneve)

Title: Word problems and geodesics in groups

Abstract: In this talk, we consider the well-studied topics of the word problem and growth of groups from the perspective of formal-language theory.

A word is a finite product of group elements, where each such element comes from a finite generating set. The word problem is the set of all words which evaluate to the group identity. The language of geodesic is the set of words which represent an element of the group with minimal length. The complexity of the word problem gives us an indication of how difficult it is to compute within a group, and the geodesic growth provides a formal-language perspective on the important topic of growth.

In this talk, we study the cases where the generating functions of these two formal languages can be written as the diagonal of a rational series. In particular, we show that this is the case for the word problem of a group which contains a direct product of a free group and an abelian group as a finite-index subgroup, and that this is the case for the language of geodesics of virtually abelian groups. Our result on the word problem answers a question in a recent paper by Pak and Soukup and generalises a result of Elder, Rechnitzer, Janse van Rensburg and Wong. Our result on the language of geodesics moves us towards a classification of groups with polynomial geodesic growth and extends results of Bridson, Burillo, Elder and Šunić.

Poster: Al@Bicocca - Bishop.pdf

17/05/2023, 3.00pm - Davide Perego (Università di Milano - Bicocca)

Title: Rationality of the Gromov boundary of a hyperbolic group

Abstract: The Gromov boundary of a hyperbolic group is a widely studied object in the field of geometric group theory. Many authors tried to provide topological approximations and recursive presentations for this boundary. After recalling some properties of hyperbolic groups and language theoretic notions, we will describe the tree of atoms. It was originally introduced by Belk, Bleak and Matucci in order to prove that hyperbolic groups embed into the rational group of asynchronous transducers. Then, we will speak about its connections with the hyperbolic world and we will see some ways to approximate and recursively present the Gromov boundary via this tree.

Poster: Al@Bicocca - Perego.pdf

29/03/2023, 3.00pm - Marco Antonio Pellegrini (Università Cattolica del Sacro Cuore)

Title: An update on the classification of the (2,3)-generated classical groups

Abstract: A group is said to be (2,3)-generated if it can be generated by an involution and an element of order 3. In this talk I will review some recent results obtained in collaboration with M.C. Tamburini about the (2,3)-generation of the finite simple orthogonal groups. These groups are the last remaining case for the complete classification of the (2,3)-generated finite simple groups.

Poster: Al@Bicocca - Pellegrini.pdf

15/03/2023, 3.00pm - Indira Chatterji (Université Côte d'Azur)

Title: CAT(0) cubical complexes: who are they and why I like them

Abstract: I will define CAT(0) cubical complexes, give a few examples and describe my favourite examples of groups acting, or not, on them.

Poster: Al@Bicocca- Chatterji.pdf

27-28 February 2023 AlBicocca Flash Workshop

27 February, 3.00pm - Cristina Acciarri (Università degli Studi di Modena e Reggio Emilia)

Title: On profinite groups with restricted centralizers of π-elements

Abstract: A group G is said to have restricted centralizers if for each g in G the centralizer C_G(g) either is finite or has finite index in G. This notion was introduced by A. Shalev, who proved that a profinite group in which all centralizers are restricted is virtually abelian, that is, has an abelian subgroup of finite index. In this talk we will consider a much more general situation, where the restrictedness condition is imposed only on π-elements of a profinite group G, for an arbitrary set of primes π.

27 February, 4.00pm - Robert F. Bailey (Memorial University of Newfoundland)

Title: Distance regular graphs with primitive automorphism groups

Abstract not avalaible

28 February, 2.00pm - Matteo Vannacci (University of the Basque Country)

Title: Iterated Wreath Products in Product Action, in search for new HJI groups

Abstract: A just infinite group is an infinite group without infinite proper quotients. A group is said to be hereditarily just infinite (HJI) if all of its finite index subgroups are just infinite. A classical classification theorem of Grigorchuk-Wilson states that a residually finite just infinite group is either: (a) a branch group or (b) an HJI group. Branch groups have been extensively studied (e.g. Grigorchuk group), but HJI groups remain a very mysterious class. In this talk, I will report on some recent work on the search for new HJI subgroups of iterated wreath products in product action.

Poster: Al@Bicocca-Flash.pdf

13/02/2023, 3.00pm - Eugenio Giannelli (University of Florence)

Title: Galois action, Sylow restriction and the McKay Conjecture

Abstract: For decades, the McKay Conjecture has been at the heart of modern representation theory of finite groups. In January 2020, Navarro and Tiep proposed a new refinement of the conjecture involving Galois action on characters and restriction of characters to Sylow subgroups. In this talk, I will describe several open problems directly connected to the McKay Conjecture and I will present some new results concerning the latest refinement proposed by Navarro and Tiep.

Poster: Al@Bicocca-Giannelli.pdf

13/12/2022, 4.30pm - James Belk (University of Glasgow)

Title: Embeddings into Finitely Presented Simple Groups

Abstract: In 1973, William Boone and Graham Higman proved that a finitely generated group G has a solvable word problem if and only if G can be embedded into a simple subgroup of a finitely presented group. They conjectured a stronger result, namely that every such group G embeds into a finitely presented simple group. This conjecture remains open after almost 50 years, but recent advances in the study of finitely presented simple groups have made it possible to verify the Boone-Higman conjecture for several large classes of groups. In this talk, I will survey results on Boone-Higman embeddings of right-angled Artin groups, countable abelian groups, contracting self-similar groups, and hyperbolic groups. This talk includes joint work with Collin Bleak, James Hyde, Francesco Matucci, and Matthew Zaremsky.

Poster: Al@Bicocca - Belk.pdf

28/11/2022, 3.00pm - Emanuele Rodaro (Politecnico di Milano)

Title: Tree automaton groups, reducible automata, and poly-context-free groups

Abstract: (Semi)groups defined by the action of Mealy transducers became very popular after the introduction of the Grigorchuk group as the first example of a group with intermediate growth. This class of groups and semigroups has deep connections with many areas of mathematics, from the theory of profinite groups to complex dynamics and theoretical computer science, and they serve as a source of examples and counterexamples for many important group theoretic problems. In this talk I will explain a construction, recently introduced in collaboration with M.Cavaleri, A.Donno, and D.D'Angeli, that allows to associate a certain automaton (semi)group to a finite simple graph. I will then focus on automaton groups associated with trees and a possible generalization called reducible automata for which we show a general structure theorem stating that all reducible automaton groups are direct limit of poly-context-free groups which are virtually subgroups of the direct product of free groups. This result partially supports a conjecture by T. Brough regarding the general structure of groups with poly-context-free word problem.

Poster: Al@Bicocca - Rodaro.pdf

14/11/2022, 3.00pm - Jan Philipp Wächter (Politecnico di Milano)

Title: Decidability Results and Open Problems for Automaton Structures

Abstract: Automaton groups and semigroups are self-similar algebraic structures generated by finite-state, letter-to-letter transducers, which in this context are usually simply called automata. The talk will give an introduction to this topic from the point of view of algebraic decision problems for such objects.

We will start with some motivation and introduce the most important objects of the area. The main part will give an overview of some known results and open problems. Here, we will discuss the finiteness problem, the order problem, some membership problems (such as the knapsack problem for groups) and the freeness problem. We will look at these problems both from a semigroup and from a group perspective and also consider special cases with regard to bounded automata (which, in the group case, are a subclass of automata characterized by the structure of their cycles).

Poster: Al@Bicocca - Wächter.pdf

25/10/2022, 5.00pm - Marialaura Noce (Università di Salerno)

Title: Hausdorff dimensions of groups acting on trees

Abstract: The concept of Hausdorff dimension was defined in the 1930s and is a way of measuring the relative size of a subset in a metric space. In the 1990s Abercrombie, Barnea and Shalev have considered the Hausdorff dimension in the setting of (countably based) profinite groups. In this talk, we will survey known results concerning the Hausdorff dimension of (the topological closure of) some distinguished families of subgroups of groups acting on trees, and we present new results and open research directions about this topic.

Poster: Al@Bicocca - Noce.pdf

17/10/2022, 3.00pm - Antonio Ioppolo (Università di Milano - Bicocca)

Title: Algebras, identities and other drugs THE SEMINAR WILL BE IN ITALIAN

Abstract: An identity is a symbolic expression involving operations and variables which is always satisfied when the variables are replaced in a given algebraic structure. I will start with a motivating example leading into the basic notions of the theory of polynomial identities in algebras. Then I will present the celebrated theorem of Amitsur and Levitzk (1950) stating that a certain standard polynomial is an identity for the algebra of square matrices. This initial combinatorial method proved to be limited until Regev introduced in 1972 a growth function measuring the size of identities. This new analytic approach, combined with techniques from ring theory, combinatorics and representation theory of groups, forms one of the current points of view of the theory. Along the way I shall try to give an idea of the recent developments of the theory when one considers associative algebras endowed with some additional structure.

Poster: Al@Bicocca - Ioppolo.pdf

## A.Y. 2021/22

20/05/2022, 4.00pm - Victor Antonio Torres Castillo (CIMAT, Mexico)

Title: Biset functors and the homotopy type of classifying spectra of saturated fusion systems

Abstract: The Segal conjecture, proved by Carlsson, revealed a connection between modular representation theory and stable homotopy theory. By the mid-nineties, Martino and Priddy used such a result to give a completely algebraic characterization of the p-local stable homotopy type of BG, where G is a finite group. In this talk, we will present a generalization of Martino-Priddy's characterization, now for saturated fusion systems over finite p-groups, under a biset functors approach and using the notion of the characteristic idempotent of a fusion system.

06/05/2022, 2.00pm - Rudradip Biswas (Tata Institute of Fundamental Research, Mumbai)

Title: Well-behaved stable categories of discrete modules over large classes of topological groups under cohomological finiteness hypotheses.

Abstract: Exploiting the recent progress made my Castellano and Weigel in studying the rational discrete modules over TDLC groups, we'll show how one can draw parallels with the behaviour of various cohomological invariants of discrete groups to construct well-behaved stable categories over TDLC groups, mirroring the constructions of a recent Mazza-Symonds paper. For this construction, the TDLC groups in question will have to satisfy certain cohomological finiteness conditions.

Poster: Al@Bicocca - Biswas.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

01/04/2022, 2.00pm - Yassine Guerboussa (The university of Ouargla, Algeria)

Title: Central Automorphisms And Related Structures

Abstract: Let G be a group. An automorphism u of G is said to be central if it induces the identity on G/Z(G). It is readily seen that every such a u induces a morphism h_u : G --> Z(G), where h_u(x)=x^{-1}x^u. In this talk, we show how the map h relates the group of the central automorphisms Aut_c(G) to an appropriate ring, and how that is useful in investigating the former, as well as in bringing out new classes of rings that are interesting in more general contexts. We shall be particularly interested to the case where G is a finite p-group.

Poster: Al@Bicocca- Guerboussa.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

25/03/2022, 2.00pm - Martin van Beek (University of Birmingham)

Title: Fusion Systems and Rank 2 Amalgams

Abstract: Saturated fusion systems capture abstract conjugacy in p-subgroups of finite groups and have found application in finite group theory, representation theory and algebraic topology. One of the ways to approach saturated fusion systems and their classification is by understanding their essential subgroups. In this talk, we classify fusion systems F in which there are two AutF (S)-invariant essential subgroups whose normalizer systems generate F. We employ the amalgam method and, as a bonus, obtain p-local characterizations of certain rank 2 group amalgams whose parabolic subgroups involve strongly p-embedded subgroups, generalizing work of Delgado and Stellmacher.

Poster: Al@Bicocca-vanBeek.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

11/03/2022, 2.00pm - Mattia Brescia (Università degli Studi di Napoli)

Title: On groups with many cyclic automorphisms

Abstract: Let G be a group. An automorphism α of G is called a cyclic automorphism if the subgroup ⟨x, x^α⟩ is cyclic for every element x of G. It can be proved that every cyclic automorphism is central, i.e. it acts trivially on the factor group G/Z(G), and that the presence of some cyclic automorphism can have a serious influence on the structure of the group. In this talk, we will describe some known properties of cyclic automorphisms of a group, providing descriptions of some classes of groups with many cyclic automorphism and presenting some open problems on the subject.

Poster: Al@Bicocca-Brescia.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

04/03/2022, 2.00pm - Alejandra Garrido (Universidad Autónoma de Madrid)

Title: A recipe for simple totally disconnected locally compact groups

Abstract: The group of automorphisms of an infinite locally finite graph can be given a totally disconnected locally compact topology with respect to which multiplication and inversion are continuous operations. In other words, it is a totally disconnected locally compact group. If this graph is, for instance, a regular tree, then its group of automorphisms is moreover (almost) simple and generated by a compact set.

In order to understand general locally compact groups, there has been a push in recent years to try to understand, and build more examples of, locally compact groups that are totally disconnected, compactly generated, simple and not discrete. As well as the automorphism group of an infinite regular tree, another typical example of this sort is the group of almost automorphisms of that tree (a.k.a. Neretin's group). This last group turns out to also be an example of a piecewise full group (a.k.a topological full group) of homeomorphisms of the Cantor set (the boundary of the tree). These piecewise full groups have been a source of new examples of finitely generated infinite simple groups. They are usually built out of certain groupoids, but in this context it is much easier to see them as coming from certain inverse semigroups of partial homeomorphisms of the Cantor set.

After introducing the necessary notions, I will report on joint ongoing work with Colin Reid and David Robertson in which we show how to build compactly generated, simple, totally disconnected locally compact groups out of certain groups, or inverse semigroups, acting on the Cantor set.

Poster: Al@Bicocca-Garrido.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

11/02/2022, 2.00pm - Matteo Vannacci (University of the Basque Country)

Title: Representations over finite fields, Probability and Zeta Functions

Abstract: In this talk we will study representation growth of a profinite group G over finite fields. On one hand, having "asymptotically few" representations over finite fields connects to (surprising) probabilistic generation properties of the completed group-ring of G. On the other hand, we can encode the number of representations of G over finite fields into a zeta-function and we will investigate how analytic properties of this zeta-function connect to properties of the original group G and its completed group-ring. This is joint work with Ged Corob-Cook and Steffen Kionke.

Poster: Al@Bicocca - Vannacci.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

28/01/2022, 2.00pm - Paola Stefanelli (Università del Salento)

Title: Set-theoretical solutions to the Yang-Baxter equation built on inverse semigroups

Abstract: The quantum Yang-Baxter equation is a fundamental equation of theoretical physics that led to the foundations of the theory of quantum groups. In the '90s, Drinfel'd posed the question of finding all set-theoretical solutions of this equation, attracting the interest of several mathematicians. In this context, in 2007, Rump traced a novel research d irect ion by introduc ing and study ing braces, a generalization of Jacobson radical rings useful for determining involutive set-theoretical solutions. This talk, based on joint work with Francesco Catino and Marzia Mazzotta, aims to introduce a new way for obtaining set-theoretical solutions involving the algebraic structure of the inverse semi-brace.

Poster: Al@Bicocca - Stefanelli.pdf

10/12/2021, 2.00pm - Davide Spriano (University of Oxford)

Title: Detecting hyperbolicity in CAT(0) spaces: from cube complexes to rank rigidity

Abstract: CAT(0) spaces form a classical and well-studied class of spaces exhibiting non-positive curvature behaviour. An important subclass of CAT(0) spaces are CAT(0) cube complexes, i.e. spaces obtained by gluing Euclidean n-cubes along faces, satisfying some additional combinatorial conditions. Given a CAT(0) cube complex, there are several techniques to construct spaces that "detect the hyperbolic behaviour" of the cube complex, but all of those techniques rely on the combinatorial structure coming from the cubes. In this talk we will present a new approach to construct such spaces that works for general CAT(0) spaces, allowing us to make progress towards the rank-rigidity conjecture for CAT(0) spaces. This is joint work with H. Petyt and A. Zalloum.

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

26/11/2021, 2.00pm - Sandro Mattarei (University of Lincoln)

Title: Graded Lie algebras of maximal class

Abstract: Interest in Lie algebras of finite coclass begun in the 1990's, when Shalev and Zelmanov discovered both similarities and striking differences with the recently proved coclass conjectures of Leedham-Green and Newman for pro-p groups. Later work by Caranti, Newman, Jurman, Vaughan-Lee led to classifications of infinite-dimensional Lie algebras of maximal class (meaning of coclass 1, also called filiform in some areas) having gradings of certain types over the positive integers. After surveying those developments I will present generalizations of some of those results, recently completed in collaboration with my former PhD students Simone Ugolini, Claudio Scarbolo and Valentina Iusa.

Poster: Al@Bicocca - Mattarei.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

12/11/2021, 2.00pm - Ilaria Colazzo (University of Exeter)

Title: Bijective set-theoretic solutions of the Pentagon Equation

Abstract: The pentagon equation appears in various contexts: for example, any finite-dimensional Hopf algebra is characterised by an invertible solution of the Pentagon Equation, or an arrow is a fusion operator for a fixed braided monoidal category if it satisfies the Pentagon Equation. This talk, based on joint work with E. Jespers and Ł. Kubat, will introduce the basic properties of set-theoretic solutions of the Pentagon Equation, present a complete description of all involutive solutions, and discuss when two involutive solutions are isomorphic.

Poster : Al@Bicocca - Colazzo.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

29/10/2021, 2.00pm - Daniele Garzoni (Tel Aviv University)

Title: On the number of conjugacy classes of a permutation group

Abstract: Let G be a subgroup of S_n. What can be said on the number of conjugacy classes of G, in terms of n?

I will review many results from the literature and give examples. I will then present an upper bound for the case where G is primitive with nonabelian socle. This states that either G belongs to explicit families of examples, or the number of conjugacy classes is smaller than n/2, and in fact, it is o(n). I will finish with a few questions. Joint work with Nick Gill.

Poster: Al@Bicocca - Garzoni.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

15/10/2021, 2.00pm - Martino Garonzi (University of Brasilia - UnB)

Title: Generating graphs and primary coverings

Abstract: I will talk about generating graphs and their connection with group coverings. I will discuss some recent results and work in progress with Fumagalli, Maróti, Gheri, Almeida. Then, I will specialize the discussion on primary coverings, i.e., coverings of elements of prime power order.

Poster: Al@Bicocca - Garonzi.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

## A.Y. 2020/21

17/06/2021, 2.00pm- Giulia Dal Verme (University of Milan - Bicocca)

Title: Grupoid actions on topological spaces and Bass-Serre theory

Abstract: The so-called Bass-Serre theory gives a complete and satisfactory descriptionof groups acting on trees via the structure theorem. We construct a Bass-Serre theory in the groupoid setting and prove a structure theorem. Groupoids are algebraic objects that behave like a group (i.e., they satisfy conditions of associativity, left and right identities and inverses) except that the multiplication operation is only partially defined. Any groupoid action without inversion of edges on a forest induces a graph of groupoids, while any graph of groupoids satisfying certain hypothesis has a canonical associated groupoid, called the fundamental groupoid, and a forest, called the Bass-Serre forest, such that the fundamental groupoid acts on the Bass-Serre forest. The structure theorem says that these processes are mutually inverse, so that graphs of groupoids "encode" groupoid actions on forests. One of the main differences between the classical setting and the groupoid one is the following: in the classical setting, given a group action without inversion on a graph, one of the ingredients used to build a graph of groups is the quotient graph given by such action; in the groupoid context, there is not a canonical graph associated to the action of a groupoid on a graph. Hence, we need to resort to the difficult notion of desingularization of a groupoid action on a graph.

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

03/06/2021, 2.00pm- Francesco Fumagalli (University of Florence)

Title: An upper bound for the nonsolvable length of a finite group in terms of its shortest law

Abstract: Every finite group G has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in G, is called the nonsolvable length λ(G) of G. In recent years several authors have investigated this invariant and its relation to other relevant parameters. E.g. it has been conjectured by Khukhro and Shumyatsky (as a particular case of a more general conjecture about non-p-solvable length) and Larsen that, if ν(G) is the length of the shortest law holding in the finite group G, the nonsolvable length of G can be bounded above by some function of ν(G). In a joint work with Felix Leinen and Orazio Puglisi we have confirmed this conjecture proving that the inequality λ(G) < ν(G) holds in every finite group G. This result is obtained as a consequence of a result about permutation representations of finite groups of fixed nonsolvable length. In this talk I will outline the main ideas behind the proof of our result.

Poster: Al@Bicocca - Fumagalli.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

28/05/2021, 11.00 am - Rossi Federico Alberto (University of Milano - Bicocca)

Title: Uniqueness of ad-invariant metrics

Abstract: Ad-invariant metrics on Lie algebras appear in the context of bi-invariant metrics on solvable Lie groups, or they are induced by the Killing form on any semisimple Lie algebra. The problem of existence of such metrics and how to construct them were studied by many authors (some examples of techniques are the double extensions and cotangent Lie algebras). In a recent paper, together with D. Conti and V. del Barco, we study the uniqueness problem of ad-invariant metrics.

In this talk, I will describe our results. First I introduce the notion of "solitary" metrics which encodes the property of uniqueness of ad-invariant metrics, and I prove that this notion depends on the underlying Lie algebra rather than on the metric itself. Then I will describe some properties of solitary Lie algebras, such as relations with complexification and cotangent construction. I will also show an equivalent notion to the solitary condition. Finally, I will state one of our main result: solitary Lie algebras are necessarily solvable. I will also recall that Lie algebras appearing in the known classifications of solvable Lie algebras admitting ad-invariant metrics are solitary, and in the end I will construct new examples of solvable Lie algebra with many inequivalent ad-invariant metrics.

Through the talk, some open problems will be presented.

Poster: Al@Bicocca - Rossi.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

06/05/2021, 2.00pm - Gareth Tracey (University of Oxford)

Title: Primitive amalgams and the Goldschmidt-Sims conjecture

Abstract: A triple of finite groups (H, M, K), usually written H > M< K, is called a primitive amalgam if M is a subgroup of both H and K, and each of the following holds:

(i) Whenever A is a normal subgroup of H contained in M, we have N_K(A) = M; and

(ii) whenever B is a normal subgroup of K contained in M, we have N_H(B) = M.

Primitive amalgams arise naturally in many different contexts across pure mathematics, from Tutte's study of vertex-transitive groups of automorphisms of finite, connected, trivalent graphs; to Thompson's classification of simple N-groups; to Sims' study of point stabilizers in primitive permutation groups, and beyond. In this talk, we will discuss some recent progress on the central conjecture from the theory of primitive amalgams, called the Goldschmidt-Sims conjecture. Joint work with László Pyber.

Poster: Al@Bicocca - Tracey.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

22/04/2021, 2.00pm - Eirini Chavli (University of Stuttgart)

Title: Real properties of generic Hecke algebras

Abstract: Iwahori Hecke algebras associated with real reflection groups appear in the study of finite reductive groups. In 1998 Broué, Malle and Rouquier generalised in a natural way the definition of these algebras to complex case, known now as generic Hecke algebras.However, some basic properties of the real case were conjectured for generic Hecke algebras. In this talk we will talk about these conjectures and their state of the art.

Poster: Al@Biccoca - Chavli.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

08/04/2021, 2.00pm- Antonio Ioppolo (University of Milano - Bicocca)

Title: Polynomial identities in associative algebras

Abstract: The main goal of this talk is to introduce the basic definitions and to present some of the most important results of the theory of polynomial identities (PI-theory) for associative algebras. When an algebra satisfies a non-trivial polynomial identity we call it a PI-algebra. The sequence of codimensions of an associative algebra over a field of characteristic zero - introduced by Regev in '72 - provides an effective way of measuring the growth of the polynomial identities satisfied by a given algebra. Regev proved that any PI-algebra has codimension sequence exponentially bounded. From that moment, the sequence of codimensions became a powerful tool in PI-theory and it has been extensively studied by several authors. In this direction, I shall present two celebrated results. The first one, proved by Kemer, characterizes those algebras having a polynomial growth of the codimension sequence. The second one is a theorem of Giambruno and Zaicev, solving in the affirmative a conjecture of Amitsur.

Poster: Al@Bicocca - Ioppolo 2021.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

25/03/2021, 2.00pm - Slobodan Tanushevski (Universidade Federal Fluminense, BRA)

Title: Frattini injective pro-p groups

Abstract: A pro-p group G is said to be Frattini-injective if distinct finitely generated subgroups of G have distinct Frattinis. Examples of Frattini-injective groups are provided by the maximal pro-p Galois groups of fields that contain a primitive p-th root of unity. I will discuss a joint work with Ilir Snopche in which we make a first attempt to systematically study Frattini-injectivity.

Poster: Al@Bicocca - Tanushevski.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

11/03/2021, 11.00 am - Mariapia Moscatiello (University of Bologna)

Title: Bases for primitive permutation groups

Abstract: The notion of a base for a permutation group is a fundamental concept in permutation group theory. The minimal cardinality of a base is called the base size of the group. Determining this invariant is a fundamental problem in permutation groups, with a long history stretching back to the nineteenth century. We will introduce the main motivations to study the base size, and we will recall some key results about this invariant. We will define the concepts of irredundant bases and we will explain the connection between these bases and the bases of minimal cardinality. We will also review some results about primitive permutation groups having all irredundant bases of the same size.

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

25/02/2021, 2.00pm - Charles Cox (University of Bristol)

Title: Spread and infinite groups

Abstract: My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are many natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, which Scott Harper recently talked about at this seminar (and mentioned several interesting questions that he and Casey Donoven posed for infinite groups in https://arxiv.org/abs/1907.05498). A group has spread if for every we can find an in such that . For any group we can say that if it has a proper quotient that is non-cyclic, then it cannot have positive spread. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread: it is also a sufficient one. But is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated. So what if we restrict ourselves to 2-generated groups? In this talk we’ll see the answer to this question. The arguments will be concrete* and accessible to a general audience.

*at the risk of ruining the punchline, we will find a 2-generated group that has every proper quotient cyclic but that has spread zero.

Poster: Al@Bicocca - Cox.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

11/02/2021, 2.00pm - Yash Lodha (Korean institute for advanced study - Seoul).

Title: Spaces of enumerated orderable groups

Abstract: An enumerated group is a group structure on the natural numbers. Given one among various notions of orderability of countable groups, we endow the class of orderable enumerated groups with a Polish topology.

In this setting, we establish a plethora of genericity results using elementary tools from Baire category theory and the Grigorchuk space of marked groups. In this talk I will describe these spaces and some of their striking features.

This is ongoing joint work with Srivatsav Kunnawalkam Elayavalli and Issac Goldbring.

Poster: Al@Bicocca - Lodha.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

28/01/2021, 2.00 pm - Albert Garreta (University of the Basque Country - Bilbao)

Title: The Diophantine problem in commutative rings and solvable groups.

Abstract: The Diophantine problem in a group or ring G is decidable if there exists an algorithm that given a finite system of equations with coefficients in G decides whether or not the system has a solution in G. I will overview recent developments that have been made in regards to this problem in the area of commutative rings and solvable groups. For large classes of such rings and groups the situation is completely clarified modulo a big conjecture in number theory. This includes the class of all finitely generated commutative rings (with or without unit), all finitely generated nilpotent groups, several polycyclic groups, and several matrix groups.

The talk is based on joint results with Alexei Miasnikov and Denis Ovchinnikov .

Poster: Al@Bicocca - Garreta.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

14/01/2021, 2.00 pm - Sabino Di Trani (Università degli Studi di Firenze)

Title: Combinatorics of Exterior Algebra, Graded Multiplicities and Generalized Exponents of Small Representations

Abstract: Let g be a simple Lie algebra over C, and consider the exterior algebra ^g as g-representations. In the talk we will give an overview of some conjectures and of many elegant results proved in the past century about the irreducible decomposition of ^g. We will focus on a Conjecture due by Reeder that generalizes the classical result about invariants in ^g to a special class of representations, called "small". Reeder conjectured that it is possible to compute the graded multiplicities in ^g of this special class of representations reducing to a "Weyl group representation" problem. We will give an idea of the strategy we used to prove the conjecture in the classical case, introducing the most relevant instruments we used and we will outline the difficulties we faced with. Finally, we will expose how our formulae can be rearranged involving the Generalized Exponents of small representations, obtaining a generalization of some classical formulae for graded multiplicities in the adjoint and little adjoint case.

Poster: Al@Bicocca - DiTrani.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

17/12/2020, 2.00 pm - Eduardo Martinez-Pedroza (University of Newfoundland, Canada)

Title: Quasi-isometric rigidity of subgroups

Abstract: A central theme in geometric group theory: what are the relations between the algebraic and geometric properties of a finitely generated group. Finitely generated groups with ``equivalent" geometries are called quasi-isometric. Let G and H be quasi-isometric finitely generated groups and let P be a subgroup of G. Is there a subgroup Q (or a collection of subgroups) of H whose left cosets coarsely reflect the geometry of the left cosets of P in G? We explore sufficient conditions on the pair (G,P) for a positive answer.

In the talk, we introduce notions of quasi-isometric pairs, and quasi-isometrically characteristic collection of subgroups. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we will describe some of them. The talk will focus on putting context to our main result and illustrate it with some applications: If G and H are finitely generated quasi-isometric groups and P is a qi-characteristic collection of subgroups of G, then there is a collection of subgroups Q of H such that (G, P) and (H,Q) are quasi-isometric pairs.

This is joint work with Jorge Luis Sanchez. UNAM, Mexico.

Poster: Al@Bicocca - Martinez-Pedroza.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

03/12/2020, 3.30 pm - Nicola Grittini (Università degli Studi di Firenze)

Title: Problems on character theory when we restrict the field of values

Abstract: Irreducible characters of rational and real values have always attracted the attention of researchers in Character theory of finite groups. One of the questions which naturally arise when these characters are studied, among the others, is whether some of the most famous results in character theory have variants involving rational or real valued characters. In fact, some of these results have such variants and, maybe not surprisingly, the variants often involve the prime number 2. An example of this fact is a theorem proved in 2007 by Dolfi, Navarro and Tiep. The theorem is a real-valued version of Ito-Michler Theorem and says that, if no real-valued irreducible character of a finite group G has even order, then the group has a normal Sylow 2-subgroup. On the other hand, it is quite difficult to work with rational and real valued characters if we consider a prime number different from 2. This suggests that, if we want to find variants of some classical results involving character fields of values and an odd prime number p, we may not consider as fields Q and R but some other fields, whose definition involves the prime p and which are equal to Q or R when p = 2. In this talk we will see two cases in which this generalization is possible, one involving rational valued and one involving real valued characters. The part involving real-valued characters has been published in a preprint and has to be considered as a work in progress, nevertheless the approach followed in the study of the problem could be interesting for many researchers in character theory.

Poster: Al@Bicocca - Grittini.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

19/11/2020, 2.00 pm - Francesco Noseda (Università Federale di Rio de Janeiro)

Title: On self-similarity of p-adic analytic pro-p groups

Abstract: A group is said to be self-similar if it admits a suitable kind of action on a regular rooted tree. Albeit it is a natural question, the study of self-similarity of p-adic analytic pro-p groups is an uncharted territory. In this talk, after recalling the basic notions, we will report on results in this direction obtained in collaboration with Ilir Snopce.

Poster: Al@Bicocca - Noseda.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

05/11/2020, 2.00pm - Scott Harper (University of Bristol)

Title: The spread of a finite group

Abstract: Many interesting and surprising results have arisen from studying generating sets for groups. For example, every finite simple group has a generating pair, and moreover Guralnick and Kantor proved that in a finite simple group every nontrivial element is contained in a generating pair. This talk will focus on recent work with Burness and Guralnick, that completely classifies the finite groups where every nontrivial element is contained in a generating pair and thus settles a 2008 conjecture of Breuer, Guralnick and Kantor. I will also give a graph theoretic interpretation of the topic, highlight how our work answers a 1975 question of Brenner and Wiegold and discuss what is known for infinite groups.

Poster: Al@Bicocca - Scott Harper

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

22/10/2020, 2.00pm - Carmine Monetta (Università di Salerno)

Title: On the exponent of the non-abelian tensor square and related constructions of finite p-groups

Abstract: If F is an operator in the class of finite groups, it is quite natural to ask whether or not it is then possible to bound the exponent of F(G) in terms of the exponent of G only, where G is a finite group. In 1991, N. Rocco introduced the operator ν which associates to every group G a certain extension of the non-abelian tensor square G ⊗ G by G ×G.

In this talk we will give an exposition of a joint work with Raimundo Bastos, Emerson de Melo and Nathália Gonçalves, where we deal with the restriction of ν to the class of finite p-groups, for p a prime. More precisely, we address the problem to determine bounds for the exponent of ν(G) and G ⊗ G when G is a finite p-group. The obtained bounds improve some existing ones and depend on the exponent of G and either on the nilpotency class or on the coclass of the finite p-group G.

Poster: Al@Bicocca - Monetta.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

08/10/2020, 2.00pm - Mima Stanojkovski (University of Bielefeld - Max Planck Institut fur Mathematik in den Naturwissenschaften )

Title: (Strong) isomorphism of p-groups and orbit counting

Abstract: The strong isomorphism classes of extensions of finite groups are parametrized by orbits of a prescribed action on the second cohomology group. We will look at these orbits in the case of extensions of a finite abelian p-group by a cyclic factor of order p. As an application, we will compute number and sizes of these orbits when the initial p-group is generated by at most 3 elements. This is joint work with Oihana Garaialde Ocaña.

Poster: Al@Bicocca - Stanojkovski.pdf

Slides: stanojkovskiHAT.pdf

## A.Y. 2019/20

16/07/2020, 6.00 pm - Pietro Gheri (Università di Firenze)

Title: Subnormalizers, probability and p-elements in finite groups

Abstract: The degree of nilpotence of a finite group is the probability that two elements in G generate a nilpotent subgroup. We show how Wielandt's subnormalizers can be used to give a classification-free proof of the fact that a nonnilpotent finite group has degree of nilpotence at most 1/2. We define a probability involving subnormalizers, sp(G), and show its relation with the number of p-elements of G, given a prime p dividing the order of G.

Poster: Al@Bicocca - Gheri.pdf

26/06/2020, 5.00 pm - Jim Belk (University of St. Andrews)

Title: Twisted Brin-Thompson Groups

Abstract: We describe a new family of finitely generated simple groups that generalize the higher-dimensional Thompson groups nV introduced by Brin. Using this family, we prove that every finitely generated group embeds quasi-isometrically into a finitely generated simple group. We also give examples of groups in this family with any given finiteness length, including groups of type F infinity. This is joint work with Matt Zaremsky.

Poster: Al@Bicocca - Belk.pdf

Recording available (to get the link, please send a request to albicocca(AT )unimib(DOT )it )

05/06/2020, 2.00 pm - MariaLaura Noce (University fo Bath)

Title: Engel conditions in groups

Abstract: The theory of Engel groups plays an important role in group theory since these groups are closely related to Burnside problems. In this talk, we survey on Engel elements and Engel groups, and we focus on their development during the last two decades, presenting new results and some open problems.

Poster: Al@Bicocca - Noce.pdf

21/05/2020, 2.30 pm - Luca Di Gravina (Università di Milano - Bicocca)

Title: The Möbius function of subgroup lattices

Abstract: Starting from the number-theoretic definition of the Möbius function, I will present its generalization to any locally finite poset with particular interest in the lattice of open subgroups of a positively finitely generated profinite group. In this case, some questions arise about the generation of groups: I focus on introducing a pair of conjectures by A. Mann and A. Lucchini. Consequently, I will explain why it is useful to find a match between the subgroup lattice of a finite group and other order structures. I will conclude by showing some possible applications to group theory.

Poster: Locandina Al@Bicocca - Di Gravina.pdf

08/05/2020, 5.00 pm - Francesco Matucci (Università di Milano - Bicocca)

Title: On Finitely Presented Groups that Contain Q

Abstract: It is a consequence of Higman's embedding theorem that the additive group Q of rational numbers can be embedded into a finitely presented group. Though Higman's proof is constructive, the resulting group presentation would be very large and unpleasant. In 1999, Martin Bridson and Pierre de la Harpe asked for an explicit and "natural" example of a finitely presented group that contains an embedded copy of Q. In this talk, we describe some solutions to the Bridson - de la Harpe problem related to Richard Thompson's groups F, T, and V. Moreover, we prove that there exists a group containing Q which is simple and has type F infinity. This is joint work with Jim Belk and James Hyde.

Poster: Al@Bicocca - Matucci.pdf

10/04/2020, 11.00 am - Valentina Grazian (Università di Milano - Bicocca)

Title: The classification of fusion systems

Abstract: Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number. Given a finite group G, it is always possible to define the saturated fusion system realized by G on one of its Sylow p-subgroups S: this is the category where the objects are the subgroups of S and the morphisms are the restrictions of conjugation maps induced by the elements of G. However, not all saturated fusion systems can be realized in this way. When this is the case, we say that the fusion system is exotic. The aim of this talk is to present two of my current research objectives: the understanding of the behavior of exotic fusion systems (in particular at odd primes) and the contribution to Aschbacher's programme of classification of saturated fusion systems at the prime 2 using the theory of localities.

Poster: Al@Bicocca - Grazian.pdf

20/02/2020, 2.30 pm - Antonio Ioppolo (Universidade Estadual de Campinas)

Title: Some PI-results in algebras with trace

Abstract: Let A be an associative algebra over a field F of characteristic zero together with its sequence of codimensions. Such a sequence was introduced by Regev in 1972 to provide an effective way of measuring the identities satisfied A by: every PI-algebra A, i.e., an algebra satisfying a non-trivial identity, has codimension sequence exponentially bounded. If V is a variety of algebras generated by A, the growth of V is defined as the growth of the codimension sequence of A. A celebrated theorem of Kemer characterizes the varieties of polynomial growth as follows: V has polynomial growth if and only if both G and UT_2 do not belong to V, where G denotes the infinite dimensional Grassmann algebra over F and UT_2 is the algebra of upper-triangular matrices over F; moreover, var(G) and var(UT_2) are the only varieties of almost polynomial growth. In this talk I will present an analogous result of Kemer’s characterisation in the context of unitary associative F-algebras that are endowed with a trace map.

Poster: Al@Bicocca - Ioppolo.pdf

04/02/2020, 2.30 pm - Pavel Zalesskii (Universitade de Brasilia)

Title: The profinite completion of 3-manifold groups

Abstract: We shall present structural results of the profinite completion of a 3-manifold group and its interrelation with the structure of . Residual properties of will be discussed.

Poster: Al@Bicocca - Zalesskii.pdf

19/12/2019, 2.30 pm - Federico W. Pasini (University of Western Ontario)

Title: The Boundary of Hyperbolic TDLC-groups

Abstract: Geometr ic Group Theory is the branch of mathematics that aims at understanding the structural properties of a given group through the geometric properties of a space on which this group acts. One of the most spectacular advance in this direction is the theory of (discrete) hyperbolic groups, i.e. groups acting nicely on a proper hyperbolic metric space, or equivalently groups whose Cayley graphs are hyperbolic. Among many other properties, a hyperbolic group admits a metrizable boundary, which turns out to provide a compactification of its Cayley graph(s). M. Bestvina and G. Mess showed a tight relationship between the cohomological properties of a hyperbolic group and of its boundary. In this talk we show that, coupling the concept of Cayley-Abels graph with the rational discrete cohomology theory developed by I. Castellano and T. Weigel, Bestvina and Mess's result can be extended to the realm of totally disconnected locally compact (= TDLC) groups. This is a joint work with Ilaria Castellano.

Poster: Al@Bicocca - Pasini.pdf

12/12/2019, 3.00 pm - Andrea Montoli (Università degli Studi di Milano)

Title: Homological properties of Schreier extensions of monoids

Abstract: In order to extend to monoids the classical equivalence between group actions and split extensions, the notion of Schreier split extension of monoids was introduced. A particular role is played by the so-called special Schreier extensions, namely those whose kernel is a group. We first give a description of Baer sums in terms of factor sets: we show that special Schreier extensions with abelian kernel correspond to equivalence classes of factor sets, as it happens for groups. Secondly, we introduce a push-forward construction for special Schreier extensions with abelian kernel in order to give an alternative, functorial description of the Baer sum, opening the way to an interpretation of the cohomology of monoids with coefficients in modules (which is a generalization of the classical Eilenberg-Mac Lane cohomology of groups) in terms of extensions. Another advantage of this functorial approach is that it can be extended to general Schreier extensions, without the assumption that the kernel is a group. We describe a classification of all such extensions in cohomological terms.

Poster: Al@Bicocca - Montoli.pdf