Algebra

Hour	Wednesday	Thursday	Friday
08h00 - 9h00	Rostislav Grigorchuk Texas A&M University- USA	Lior Bary Soroker Tel Aviv University- Israel	Ashot Minasyan University of Southampton- UK
09h30 - 10h30	Dessislava Kochloukova Unicamp	Angel Del Rio Universidad de Murcia- Spain	Ilir Snopce UFRJ
10h30-11h00	coffee break
11h00 - 12h00	Ilaria Castellano University of Milano-Bicocca- Italy	Wolfgang Herfort Vienna University of Technology- Vienna	Andrei Jaikin Autonomous University of Madrid- Spain
12h00 - 14h00	Lunch
14h00 - 15h00	Alex Lubotzky Weizmann Institute of Science - Israel	Dan Segal University of Oxford - UK	Alexandre Zalesski University of East Anglia - UK
15h10 -16h00	John MacQuarrie UFMG	Marco Boggi UFMG	Slobodan Tanushevski UFF
16h00 - 16h30	coffee break
16h30 - 17h20	Henrique Souza Autonomous University of Madrid- Spain	Carmine Monetta Università degli Studi di Salerno -Italy	PZ Press Conference

Talks

Rostislav I. Grigorchuk (Texas A&M University- USA) On subgroup structure of groups of branch type.
PDF | Abstract

Groups of branch type (abstract or profinite) is a remarkable class of groups full of groups with unusual properties and related to the numerous branches of mathematics. Some of such groups bear intriguing names like Hanoi Towers Groups, Basilica, IMG(z^2+i), .... This class contains groups of intermediate growth (between polynomial and exponential), amenable but not elementary amenable groups, groups of Burnside type (i.e. infinite finitely generated torsion groups), just-infinite groups, finitely constrained groups etc.
Also the groups of branch type have unusual structure of the lattice of subgroups. In my talk first, I will give a panorama of known results about subgroups of branch (and weakly branch) groups starting from congruence subgroup and subgroup separability (or LERF) properties and then focusing on maximal and weakly maximal subgroups. Then I will define the block structure for finitely generated subgroups of self-similar branch groups and formulate a few results. The material of the talk will be based on the numerous results of the speaker and his coouthers, including the jubilee Pavel Zalesskii.

Dessislava Kochloukova (Unicamp)Higher dimensional algebraic fiberings .
PDF | Abstract

We will discuss a pro-p version of results of higher dimensional algebraic fiberings. The case of abstrac groups is a joint work with Stefano Vidussi. Both preprints are posted on arxiv. As a corollary we will discuss some results of incoherence.

Ilaria Castellano (University of Milano-Bicocca- Italy) A pro-p version of Sela's accessibility
PDF | Abstract

Accessibility of splittings over finite groups (i.e., as a graph of groups with finite edge groups) was studied by Dunwoody who proved that finitely presented groups are accessible but found an example of an inaccessible finitely generated group. This initiated naturally a search for a kind of accessibility that holds for finitely generated groups. The breakthrough in this direction is due to Sela who proved $k$-acylindrical accessiblity for finitely generated groups: accessibility provided the stabilizer of any segment of length $k$ of the group acting on its standard tree is trivial for some $k$.
In general finitely generated pro-$p$ groups are not accessible, as shown by Wilkes, and it is an open question whether finitely presented are. In this talk we will present the pro-$p$ version of Sela’s result and, time permitting, we will provide an application for it.
Joint work with Pavel Zalesskii.

Alex Lubotzky (Weizmann Institute of Science - Israel) Stability and testability of permutations' equations.
PDF | Abstract

We study a new direction of research in "property testing": Permutation equations. Let A and B be two permutations in Sym(n) that "almost commute"- are they a small deformation of permutations that truly commute? More generally, if R is a system of word-equations in variables X=x_1,....,x_d and A_1,...., A_d are permutations that are almost a solution; are they near a true solution? It turns out that the answer to this question depends only on the group presented by the generators X and relations R. This leads to the notions of "stable groups" and "testable groups" and calls for some group theoretic methods for answering these questions. We will present a few results and methods which were developed in recent years to check whether a group is stable\testable ( e.g., using IRS's= invariant random subgroups). We will also describe the connection of this subject with locally testable codes as well as with the long-standing problem of whether every group is sofic.

John MacQuarrie ( UFMG ) Block Theory for profinite groups
PDF | PDF | Abstract

If G is a finite group and k is a field, the group algebra kG (as with any finite dimensional algebra) can be decomposed as a direct product of indecomposable algebras, called the blocks of G over k. Great progress in the representation theory of finite groups has been made by studying the kG-modules “one block at a time” (this approach is “block theory’). I will discuss distinct projects, with Ricardo Franquiz Flores (UNIFEI) and with Peter Symonds (University of Manchester) wherein we develop a block theory for profinite groups, trying to emphasise how remarkably robust and well-behaved the theory seems to be.

Henrique Souza( Autonomous University of Madrid - Spain ) Completed group algebras of free-by-Zp pro-p groups
PDF | Abstract

The class of finitely generated free-by-Zp pro-p groups comprise an important family of pro-p groups of cohomological dimension at most 2, amongst which one can find free pro-p and Dëmushkin groups. In this talk I will present (in joint work with A. Jaikin) how those groups are virtually mild in the sense of Labute and how they are characterized by possessing a very particular type of presentation. Moreover, I will also explain how a strong control of the restricted Lie algebra structure obtained by these presentations allows us to virtually construct an explicit embedding of the completed Fp-group algebra into a division ring . As a corollary, one obtains a virtual solution to the Atiyah conjecture for this class of groups.

Lior Bary-Soroker ( Tel Aviv University- Israell ) Probabilistic Galois Theory.
PDF | Abstract

The talk concerns the following question: Given a random polynomial of degree n with integer coefficients, how small is the probability that its Galois group is the alternating group? This question goes back at least to the work of van der Waerden in 1936, and has been studied along the years since then. Recently, Bhargava made a breakthrough and showed that this probability is not asymptotically bigger than the probability of being reducible. However, one expects the probability to be much smaller; We will present a conjectured order of magnitude for the probability and two recent pieces of evidence for the conjecture. Based on joint works with Ben-Porath and Matei, and with Woo.

Angel del Río (Universidad de Murcia- Spain) Recent developments in the isomorphism problem for group rings.
PDF | Abstract

RECENT DEVELOPMENTS IN THE ISOMORPHISM PROBLEM FOR GROUP RINGS
ÁNGEL DEL RIO
If R is a ring and G is a group then RG denotes the group ring of G with coef- ficients in R. The Isomorphism Problem, for R a commutative ring, asks whether the isomorphism type of RG as R-algebra determines the isomorphism type of G. The special case where R is field of characteristic p and G is a finite p-group is known as the Modular Isomorphism Problem. While a negative solution for the Modular Isomorphism Problem in characteristic 2 has been found recently [2], for odd characteristic it is still an open question. The example in [2] is 2-generated and cyclic-by-abelian. However, we will present evidence that a similar example cannot be constructed in odd characteristic [3]. We will present also some positive results on the Isomorphism Problem for ra- tional group algebras.
References
[1] A`.Garc ́ıa-Bla ́zquez,andA ́.delR ́ıo,Aclassificationofmetacyclicgroupsbygroupinvariants, http://arxiv.org/abs/2301.08683.
[2] D. Garc ́ıa-Lucas, L. Margolis, and A ́. del R ́ıo, Non-isomorphic 2-groups with isomorphic modular group algebras, Journal fur die Reine und Angewandte Mathematik, 783 (2022) 269–274.
[3] D. Garc ́ıa-Lucas, A ́. del R ́ıo, and M. Stanojkowsky, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebras and Representation Theory.
doi.org/10.1007/s10468-022-10182-x.

Wolfgang N. Herfort (Vienna University of Technology- Vienna) Free profinite products of prosoluble groups
PDF | Abstract

Let C be a variety of finite groups and p a prime. Then the free pro-C product G=\amalg_{i\in I} A_i for pro-p groups A_i splits as a semidirect product of the pro-p free product of the A_i and O_p(G). There are counter-examples limiting possible generalizations.
joint work with K Ersoy

Dan Segal (University of Oxford - UK) Profinite groups and logic
PDF | Abstract

Which profinite groups can be specified by first-order sentences, or by finitely many such sentences? Brief survey of some answers and some more specific questions.

Marco Boggi ( UFMG ) Automorphisms of procongruence mapping class groups
PDF | Abstract

For $S = S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, with $χ_p(S)= 2- 2g- n<0$, let $\Gamma(S)$ be the mapping class group of $S$ and $\widehat{\Gamma}(S)$ and $\tilde{\Gamma}(S)$, respectively, its profinite and its congruence completion. The latter is the image of the natural representation $\tilde{\Gamma}(S) \rightarrow Out({\widehat{\pi_1}}(S))$, where $\widehat{\pi_1}}(S)$ is the profinite completion of the fundamental group of the surface $S$. Let $Out^{I_0}(\tilde{\Gamma}(S))$ be the group of outer automorphisms of $\tilde{\Gamma}(S)$, which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist and put $d(S)= 3g - 3+ n$. In this talk, I will discuss the theory developed in the paper [1]. The main result is that, for $d(S) > 1$, there is a natural isomorphism: $Out^{I_0}(\tilde{\Gamma}(S))\cong \wildehat{GT}$, where $GT$ is the profinite Grothendieck-Teichmüller group. We will actually prove a slightly stronger result which implies that the automorphism group of the procongruence Grothendieck-Teichmü̈ller tower is also isomorphic to $GT$.
References
[1] M. Boggi. Automorphisms of procongruence mapping class groups. https://arxiv.org/abs/2011.15075 (2022).

Carmine Monetta ( Università degli Studi di Salerno -Italy ) The nilpotency of groups with isomorphic non-commuting graphs..
PDF | Abstract

Given a finite group $G$, one can consider a graph $\Gamma_G$ associated with $G$ which encodes certain group properties of $G$. Such an approach has been extensively studied in the last decades, mainly to determine structure description of $G$ investigating the invariants of $\Gamma_G$.
A natural question in this research line is to understand if a graph isomorphism - which is clearly a weaker relation than a group isomorphism - may or may not preserve specific properties of a group. More precisely, we are interested in the following question. Assume that $G$ and $H$ are finite groups with isomorphic graphs $\Gamma_G \cong \Gamma_H$. If $G$ is nilpotent, is it true that $H$ is nilpotent as well?
Of course the hardness of the problem, as well as the answer, change depending on the graph choice. In this talk we report on this problem, clarifying that the situation is not so easy in general and that in some cases the problem remains still open.

Ashot Minasyan (University of Southampton- UK) The Ribes-Zalesskii property for graphs of free groups with cyclic edges.
PDF | Abstract

Let G be a group and let n be a natural number. The group G is said to have the Ribes-Zalesskii property RZ_n if for arbitrary finitely generated subgroups H_1,...,H_n of G, the product H_1...H_n is closed in the profinite topology on G.
Thus property RZ_1 means that G is subgroup separable (LERF) and property RZ_2 means that G is double coset separable. If G satisfies RZ_n for all natural numbers n then G is called product separable.
In 1993 Ribes and Zalesskii proved that free groups are product separable. This confirmed a conjecture of Pin and Reutenauer, that was motivated by questions from Semigroup Theory. In my talk I will discuss recent joint work with Lawk Mineh, where we investigated product separability for groups hyperbolic relative to families of product separable subgroups. In particular we show that limit groups, finitely generated Kleinian groups and balanced fundamental groups of finite graphs of free groups with cyclic edge groups are all product separable.

Ilir Snopce (UFRJ- Brazil) Maximal pro-p Galois groups.
PDF | Abstract

Given a pro-p group G, let Φ(G) be the Frattini subgroup of G. We call a pro-p group G Frattini-resistant if for all finitely generated subgroups H and K of G, Φ(H) ≤ Φ(K) implies H ≤ K. I will talk about Frattini-resistant pro-p groups and their relation with maximal pro-p Galois groups. Moreover, I will discuss the following question: for which simplicial graphs the pro-p completion of the corresponding right-angled Artin group occurs as the maximal pro-p Galois group of some field containing a primitive pth root of unity? This talk is based on joint work with Slobodan Tanushevski and joint work with Pavel Zalesski.

Andrei Jaikin Zapirain (Autonomous University of Madrid- Spain) On recent advances on the profinite rigidity of free and surface groups.
PDF | Abstract

I will discuss recent results about the Profinite Rigidity of free and surface groups. In particular I will explain the proofs of the following two results:
1) Let $\phi: F\to G$ be an injective map from a free group to a residually finite group inducing an isomorphism of profinite completions. Then $\phi$ is an isomorphism.
2) Let $G$ be a residually finite one-relator group having the same profinite completion as a surface group. Then $G$ is surface. (joint with Ismael Morales)

Alexandre Zalesski (University of East Anglia - UK) Eigenvalue problems in group representation theory
PDF | Abstract

Studying properties of individual group elements in group representations is important for various applications. In this talk I focus on studying the degrees of minimal polynomials of a group element and the existence of eigenvalue 1. After a short overview of this research area, I shall state some open problems and results obtained.

Slobodan Tanusevski ( UFF ) Test elements and retracts in discrete groups and pro-p groups.
PDF | Abstract

An element $g$ of a group $G$ is called a test element if every endomorphism of $G$ that fixes $g$ is an automorphism. The notion of a test element was introduced by Vladimir Shpilrain as a way of organizing a plethora of hitherto scattered results into a general conceptual framework.
In many groups, the set of test elements coincides with the complement of the union of all proper retracts of the group. This complementary relation between retracts and test elements was first discovered by Edward Turner.
In this talk, I will discuss test elements and retracts in pro-p groups. I hope to convince you that test elements, and several other related concepts, have a significant role to play in the theory of pro-p groups. In addition, I will present several results on retracts and test elements of discrete groups obtained by means of pro-p techniques. The talk is based on several joint papers with Ilir Snopce and a joint paper with Ilir and Pavel Zalesski.

PZ press conference

WORKSHOP ON COMBINATORIAL THEORY OF PROFINITE GROUPS AND APPLICATIONS

Workshop in honour of Pavel Zalesskii's 60th birthday
15 - 17 march, Brazil

ABOUT THE WORKSHOP

Scientific Committee

Local Committee

Invited speakers

REGISTRATION

SCHEDULE

Schedule

Talks

SPONSORS

CONTACT US

BOOK PHOTO

WORKSHOP ON COMBINATORIAL THEORY OF PROFINITE GROUPS AND APPLICATIONS

Workshop in honour of Pavel Zalesskii's 60th birthday15 - 17 march, Brazil

ABOUT THE WORKSHOP

Scientific Committee

Local Committee

Invited speakers

REGISTRATION

SCHEDULE

Schedule

Talks

SPONSORS

CONTACT US

BOOK PHOTO

Workshop in honour of Pavel Zalesskii's 60th birthday
15 - 17 march, Brazil