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Pavel Turek: On stable modular plethysms of the natural module of SL2(𝔽p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(𝔽p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(𝔽p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(𝔽p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

Umberto Zannier: Bounded generation in linear groups and exponential parametrizations

31st December, 2022

In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of 'bounded generation' in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on 'anisotropic groups' (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are 'sparse'. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.

Pavel Zalesski: Combinatorial theory of pro-p groups

15th December, 2022

Free products with amalgamation and HNN-extensions are two main constructions of combinatorial group theory. I shall discuss these two constructions in the category of pro-p groups, presenting results on splittings of pro-p groups as an amalgamated free pro-p product or a pro-p HNN-extension and relating them with pro-p version of Bass-Serre's theory of groups acting on trees. I shall also compare the pro-p results with similar results for abstract groups.

Thomas Koberda: First-order rigidity of homeomorphism groups of manifolds

9th December, 2022

I will discuss some aspects of the first-order theory of homeomorphism groups of connected manifolds. The main result is as follows. Let M be a compact, connected manifold. There is a sentence S(M) in the language of groups such that if N is an arbitrary manifold and the homeomorphism group of N models S(M) then N is homeomorphic to M. This resolves a conjecture of Rubin from the 1980s. I will illustrate some of the ingredients of the proof, including an interpretation of second order arithmetic in the theory of homeomorphism groups of manifolds.

Agatha Atkarskaya: Introduction to group-like small cancellation theory for rings

18th November, 2022

The structure of small cancellation groups is well known. Тhey are widely used in construction of groups with unusual properties (for example Burnside groups and Tarskii monster). We were interested in developing a similar theory for rings. However, such theory meets significant difficulties because, unlike groups, rings have two operations: addition and multiplication. I will speak about small cancellation conditions for rings that we introduced. These conditions provide the desired properties. I will discuss our way towards these conditions, examples and possible applications of small cancellation rings.

Jeroen Schillewaert: Constructing highly regular expanders from hyperbolic Coxeter groups

17th November, 2022

Given a string Coxeter system (W,S), we construct highly regular quotients of the 1-skeleton of its universal polytope P, which form an infinite family of expander graphs when (W,S) is indefinite and P has finite vertex links. The regularity of the graphs in this family depends on the Coxeter diagram of (W,S). The expansion stems from superapproximation applied to (W,S). This construction is also extended to cover Wythoffian polytopes. As a direct application, we obtain several notable families of expander graphs with high levels of regularity, answering in particular a question posed by Chapman, Linial and Peled positively.

This talk is based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.

This video was produced by the Sydney Mathematical Research Institute, as part of their SMRI seminar series.

Giada Volpato: On the restriction of a character of Sn to a Sylow p-subgroup

15th November, 2022

The relevance of the McKay Conjecture in the representation theory of finite groups has led to investigate how irreducible characters decompose when restricted to Sylow p-subgroups. In this talk we will focus on the symmetric groups. Since the linear constituents of the restriction to a Sylow p-subgroup has been studied a lot by E. Giannelli and S. Law, we will concentrate on constituents of higher degree. In particular, we will describe the set of the irreducible characters which allow a constituent of a fixed degree, separating the cases of p being odd and p=2.

Yifan Jing: Measure Growth in Compact Simple Lie Groups

8th November, 2022

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| ≫ min ( |A|1+c, |G| ). In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact simple Lie group G, then μ(AAA) > min ( (3+c)μ(A), 1 ), where μ is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.

Pavel Shumyatsky: Commuting probability for subgroups of a finite group

4th November, 2022

If K is a subgroup of a finite group G, the probability that an element of G commutes with an element of K is denoted by Pr(K,G). The probability that two randomly chosen elements of G commute is denoted by Pr(G). A well-known theorem, due to P. M. Neumann, says that if G is a finite group such that Pr(G) ≥ ε, then G has a nilpotent normal subgroup T of class at most 2 such that both the index [G:T] and the order |[T,T]| are ε-bounded.

In the talk we will discuss a stronger version of Neumann's theorem: if K is a subgroup of G such that Pr(K,G) ≥ ε, then there is a normal subgroup T ≤ G and a subgroup B ≤ K such that the indices [G:T] and [K:B] and the order of the commutator subgroup [T,B] are ε-bounded.

We will also discuss a number of corollaries of this result. A typical application is that if in the above theorem K is the generalized Fitting subgroup F*(G), then G has a class-2 nilpotent normal subgroup R such that both the index [G:R] and the order of the commutator subgroup [R,R] are ε-bounded.

Shai Evra: Optimal strong approximation and the Sarnak-Xue density hypothesis

3rd November, 2022

It is a classical result that the modulo map from SL2(ℤ) to SL2(ℤ/qℤ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL2(ℤ/qℤ) has a lift in SL2(ℤ) with coefficients of size at most qe? A simple pigeonhole principle shows that e is strictly greater than 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL2(ℤ). The proof relies on a density theorem of the Ramanujan conjecture for SL2(ℤ). In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands programme.

Alexander Hulpke: Constructing Perfect Groups

2nd November, 2022

The construction of perfect groups of a given order can be considered as the prototype of the construction of insoluble groups of a given order. I will describe a recent project to enumerate, up to isomorphism, the perfect groups of order up to 2⋅106. It crucially relies on new tools for calculating cohomology, as well as improved implementations for isomorphism test. This work extends results of Holt and Plesken from 1989 and illustrates the scope of algorithmic improvements over the past decades.

Laurent Bartholdi: Dimension series and homotopy groups of spheres

28th October, 2022

The lower central series of a group G is defined by γ1=G and γn = [G,γn-1]. The 'dimension series', introduced by Magnus, is defined using the group algebra over the integers:

δn = { g : g-1 belongs to the nth power of the augmentation ideal }.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δn ≥ γn, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ4 / γ4 cyclic of order 2. On the positive side, Sjogren showed that δn / γn is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proved by Gupta) that only 2-torsion may occur.

In joint work with Roman Mikhailov, we prove however that every torsion abelian group may occur as a quotient δn / γn; this proves that Sjogren's result is essentially optimal.

Even more interestingly, we show that this problem is intimately connected to the homotopy groups πn(Sm) of spheres; more precisely, the quotient δn / γn is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group π2p(S2) due to Serre.

Chen Meiri: Word Width in Higher Rank Arithmetic Groups

18th October, 2022

A word on d letters is an element of the free group of rank d, say, with basis x1,…,xd. Given a word w=w(x1,…,xd) on d letters, for every group G, there is a word map w:Gd→ G given by substituting the xi with elements of G. We say that a word w has a finite width n in the group G if any element in the subgroup generated by w(G) is a product of at most n element of w(G) or their inverses. In this talk, I will survey results about word width in several families of groups and then restrict the focus to the family of higher rank arithmetic groups. I will present a conjecture about word width in higher-rank arithmetic groups and explain some consequences, most notably, to the Congruence Subgroup Problem.

Haralampos Geranios: On self-extensions of irreducible modules for symmetric groups

11th October, 2022

We work in the context of the modular representation theory of the symmetric groups. A long-standing conjecture, from the late 80s, suggests that there are no (non-trivial) self-extensions of irreducible modules over fields of odd characteristic. In this talk we will highlight several new positive results on this conjecture.

Nikolay Nikolov: On conjugacy classes of profinite groups

10th October, 2022

It is well-known that the number of conjugacy classes of a finite group G tends to infinity as the size of G tends to infinity. There is no such result for a general infinite group. In this talk I will discuss the situation when G is a profinite group and show that the number of conjugacy of G is then uncountable unless G is finite. The proof depends on many classical results on finite groups and in particular the classification of the finite simple groups.

Matthew Conder: Discrete 2-generator subgroups of PSL2(ℚp)

10th October, 2022

Discrete 2-generator subgroups of PSL2(ℝ) have been extensively studied by investigating their action by Möbius transformations on the hyperbolic plane. Due to work of Gilman, Rosenberger, Purzitsky and many others, there is a complete classification of such groups by isomorphism type, and an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℝ) is discrete.

Here we completely classify discrete 2-generator subgroups of PSL2(ℚp) over the p-adic numbers ℚp by studying their action by isometries on the corresponding Bruhat-Tits tree. We give an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℚp) is discrete, and discuss how this can be used to decide whether or not a 2-generator subgroup of SL2(ℚp) is dense.

Volker Diekert: Decidability of membership problems for 2×2 matrices over ℚ

7th October, 2022

We consider membership problems in matrix semigroups. Using symbolic algorithms on words and finite automata, we prove various new decidability results for 2×2 matrices over ℚ. For that, we introduce the concept of flat rational sets: if M is a monoid and N is a submonoid, then flat rational sets of M over N are finite unions of the form L0g1L1⋯gtLt where all Li are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL2(ℚ) over GL2(ℤ) is decidable (in singly exponential time). It is possible that such a strong decidability result cannot be pushed any further for groups sitting between GL2(ℤ) and GL2(ℚ).

We also show a dichotomy for non-trivial group extension of GL2(ℤ) in GL2(ℚ): if G is a f.g. group such that GL2(ℤ) < G ≤ GL2(ℚ), then either G ≅ GL2(ℤ) × ℤk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1,q), with q ≥ 2, of infinite index. In the first case of the dichotomy the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem for rational subsets in G is open.

Our result improves various natural decidability results for 2×2 matrices with rational entries, and it also supports them with concrete complexity bounds for the first time.

Eric Freden: Aspects of growth in Baumslag-Solitar groups

7th October, 2022

In 1997, Grigorchuk and de la Harpe suggested computing the growth series for the Baumslag-Solitar group BS(2,3). After 25 years, this is still an open problem. In fact, the growth of only the solvable groups BS(1,n) and automatic groups BS(n,n) are known. In this talk I will review what has since been discovered about these remarkable groups and conclude with new unpublished results concerning the exponents of growth for the subfamily BS(2,2n).

Kane Townsend: Hyperbolic groups with k-geodetic Cayley graphs

7th October, 2022

A locally-finite simple connected graph is said to be k-geodetic for some k ≥ 1, if there are at most k distinct geodesics between any two vertices of the graph. We investigate the properties of hyperbolic groups with k-geodetic Cayley graphs. To begin, we show that k-geodetic graphs cannot have a "ladder-like" geodesic structure with unbounded length. Using this bound, we generalize a well-known result of Papasoglu that states hyperbolic groups with 1-geodetic Cayley graphs are virtually free. We then investigate which elements of the hyperbolic group with k-geodetic Cayley graph commute with a given infinite order element.