Gong Show

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Gong Show Session 1: Permutation and symmetric groups

Jessica Anzanello: Nearly fixed-point-free elements in permutation groups with two orbits

A classical theorem of Jordan states that every non-trivial finite transitive group contains a derangement, that is, an element with no fixed points. A deep result of Fein, Kantor, and Schacher shows that one can even find such an element of prime-power order. In this talk, I will discuss a natural extension of this result to permutation groups G with two orbits. In this setting, we show that either G contains a derangement, or it contains an element of prime-power order with exactly one fixed point. As a corollary, we obtain that if the two orbits have size a and b, with a + b > 2 and (a, b − 1) = (a − 1, b) = 1, then G has a derangement. This confirms and generalizes a conjecture of Ellis and Harper. This is joint work with Sean Eberhard.
Adrian Beker: The Rényi entropy of the order of a random permutation

The study of the order of a random permutation is a fundamental problem in probabilistic group theory. Its global behaviour is well understood: a classical result of Erdős and Turán shows that the logarithm of the order is asymptotically normally distributed. We consider two local questions: what is the most probable order of a random permutation of {1, . . . , n}, and with what probability does it occur? What is the probability that two independent random permutations have equal orders? The former goes back to work of Erdős and Turán, while the latter was recently studied by Acan, Burnette, Eberhard, Schmutz and Thomas. We give an essentially complete answer to these questions by analysing the associated concept of Rényi entropy. Our results are quantitatively optimal and reveal a tight connection between the asymptotic behaviour of the quantities in question and number-theoretic properties of n.
Ohad Sheinfeld: Improved covering results and intersection theorems in symmetric groups, via hypercontractivity

In this talk, we present two seemingly unrelated results on the symmetric group S_n. A well-known problem asks for a characterization of subsets A of A_n whose square A² covers the entire alternating group A_n. We show that if A is a conjugacy class of density at least exp(−n^2/5 − ε), then A_n = A². This improves upon a seminal result by Larsen and Shalev, who obtained the same result with 1/4 in the double exponent.

The second problem is an analogue of the Erdős–Sós forbidden intersection problem for families of permutations. A family F of permutations is called non-(t − 1)-intersecting if any two permutations in F do not agree on exactly (t − 1) values. We prove that for t < √n / log(n), the maximal size of a non-(t − 1)-intersecting family is (n − t)!, which improves upon the result of Kupavskii and Zakharov, who proved the same for t = Õ(n^1/3).

Both results can be interpreted as bounds on the size of independent sets in certain normal Cayley graphs over symmetric groups. The primary tool we use in both proofs to establish these bounds are the recently obtained hypercontractive inequalities for global functions, developed by Keller, Lifshitz, and Marcus, as well as by Keevash and Lifshitz. The talk is based on joint work with Nathan Keller and Noam Lifshitz.

Gong Show Session 2: Lie groups and algebraic groups

Arunava Mandal: Cartan subgroups and strong approximate lattices in Lie groups

Cartan subgroups are classical and fundamental objects in Lie theory, playing an important role in the structure and representation theory of Lie groups. In this talk, we discuss some new results concerning the structure of Cartan subgroups in connected Lie groups. On the other hand, approximate lattices, which generalise the notion of lattices, have recently emerged as an active area of research. We will explain connections between Cartan subgroups and strong approximate lattices in linear semisimple Lie groups without compact factors. In particular, our results extend classical theorems of George Mostow and partially generalise results of Gopal Prasad and M. S. Raghunathan. This is joint work with Sashank Vikram Singh.
Lal Bahadur Sahu: Hypergeometric groups and a question of Sarnak

In this talk, we will begin with the foundational framework of Beukers and Heckman, which classifies the Zariski closures of hypergeometric groups. Specifically, we will discuss the criteria for arithmeticity developed by Singh and Venkataramana and contrast this with the results of Brav and Thomas demonstrating the thinness of specific Calabi–Yau hypergeometric groups. If time permits, we will review the progress on the question asked by Sarnak to determine the arithmeticity and thinness of the hypergeometric groups so far.
Dávid Szabó: Toric decomposition in algebraic groups

We study an analogue of the tiling problem for finite groups in the setting of linear algebraic groups over algebraically closed fields using maximal tori. For G = PSL_n, we construct concrete maximal tori T₁, ..., T_n+1, such that the product T₁...T_n+1 = {t₁...t_n+1 : t_i ∈ T_i} is Zariski dense in G, and every g ∈ T₁...T_n+1 admits a unique factorisation g = t₁...t_n+1, where t_i ∈ T_i. More generally, if G is a connected reductive algebraic group and k ≤ dim(G) / rank(G), we show the existence of maximal tori T₁, ..., T_k of G such that generic elements g ∈ T₁...T_k admit only finitely many factorisations of the form g = t₁...t_k, where t_i ∈ T_i. If time permits, we mention several similar open questions regarding toric decompositions in groups. This is joint work with Endre Szabó.
Ken Vandermeersch: Dehn functions of nilpotent groups

I am interested in Dehn functions of finitely generated nilpotent groups, and in particular in how sharp bounds can be extracted from their Lie-algebraic structure. One useful lower-bound invariant is the centralized Dehn function, which is often much easier to compute than the ordinary Dehn function and can be computed using the cohomology of the associated Mal’cev Lie algebra.

In this short talk, I will explain the idea behind this invariant and discuss the question of how far it can be from the ordinary Dehn function. Although the centralized Dehn function gives sharp bounds in many classical examples, it is not known in general how large this gap can be. Recent examples where the polynomial exponents differ show only an exponent gap of 1, but it is natural to ask whether this gap can be made arbitrarily large.

Gong Show Session 3: Cayley graphs, expansion, and isoperimetry

Marco Barbieri: Strong convergence of random permutational representations

A family of d-regular graphs is called an expander family if their adjacency matrices exhibit a spectral gap between the trivial eigenvalue d and the rest of the spectrum. This seemingly simple condition has far-reaching consequences, from rapid mixing of random walks to strong metric properties. On a seemingly different note, a sequence of random matrices is said to strongly converge to an operator if, with high probability, the norms of these matrices converge to the operator norm of the limiting object. This notion has recently found powerful applications in expansion problems, both in graph theory and in hyperbolic geometry. In this short talk, we will explore the connection between these two notions, and we will show how strong convergence of random permutational representations of symmetric groups leads to probabilistic constructions of expanders. This is joint work with Urban Jezernik.
Nathan Deloire: Isoperimetry of diagonal products of permutational wreath products

While a construction of groups with prescribed isoperimetric behaviour is known in the exponential growth setting, much less is understood for groups of intermediate growth.

In 2021, Jérémie Brieussel and Tianyi Zheng constructed finitely generated groups with prescribed isoperimetry for a large class of isoperimetric profiles, or equivalently return probabilities. These groups are constructed using the notion of “diagonal product” of marked wreath products Γ_m ≀ Z, with (Γ_m)_{m ∈ N} a sequence of finite groups, and yield groups of exponential growth. The key step of the proof is the evaluation of the asymptotic isoperimetry of such groups.

In my research, I am adapting this idea in order to construct intermediate growth groups with prescribed isoperimetry by studying the asymptotic isoperimetry of the diagonal products of groups of intermediate growth. The groups of which I am studying the diagonal product are permutational wreath products Γ_m ≀_X G_ω, with (Γ_m)_{m ∈ N} a sequence of finite groups and G_ω a Grigorchuk group, and whose intermediate growth was proved in 2014 by Bartholdi and Erschler.
Cosmas Kravaris: Do the symmetric groups embed into the Banach space L₁?

Kassabov’s theorem states that the symmetric groups admit bounded size generating sets whose Cayley graphs form expanders. When the generating set is a cycle and a transposition, it is easy to check that the Cayley graph is not an expander. By a classical argument of Gromov, expander graphs do not embed coarsely into L₁. On the other hand, we show that the Cayley graph generated by a cycle and a transposition embeds into L₁ with bi-Lipschitz distortion < 1000. In sharp contrast to the study of infinite groups, we see that the large-scale metric geometry of a finite group depends heavily on the choice of generators.
Elena Maini: Diameter bounds for Cayley graphs of arbitrary finite groups

Investigating the growth of a finite group involves looking at its Cayley graphs. For example, it is interesting to estimate the maximum diameter of a connected Cayley graph of a finite group G, which is usually referred to as the diameter of G. I will present a new result stating that, given an arbitrary finite group G, the diameter of G can be controlled provided that the normal abelian sections of G have small exponent and the composition factors of G have small diameter. I will then discuss some surprising consequences of this fact. This is joint work with Sean Eberhard, Luca Sabatini, and Gareth Tracey.
Luca Sabatini: Cayley graphs of quasirandom groups

Building on the recent work of Golsefidy–Srinivas, we present new expansion properties of abstract quasirandom groups.
Yotam Shomroni: Word measures on finite groups and expansion of random Schreier graphs

Understanding word measures on a group — specifically, giving upper bounds on word measures of non-trivial irreducible characters — is a widely used tool for proving spectral expansion of random Schreier graphs of certain families of finite groups, such as the symmetric groups. I work on word measures, their relations to the subgroup structure of the free group, and their applications to expansion.

Gong Show Session 4: Harmonic analysis, combinatorial, and geometric group theory

Soumyadeb Samanta: Lipschitz harmonic functions and coarse harmonic coordinates on groups of polynomial growth

We shall discuss the theory of harmonic functions on finitely generated groups, with emphasis on linear growth harmonic functions, especially the Lipschitz harmonic functions and associated results. We shall see that on finitely generated nilpotent groups, for any adapted, smooth, Abelian-centered probability measure μ, every Lipschitz-harmonic function is of the form f(x) = c + ϕ(x), ϕ ∈ Hom(G, C). For any finite generating set S of G this yields a canonical identification LHF(G, μ)/C ≅ Hom(G, C), ||∇_Sf||_∞ = max_s∈S|ϕ(s)|, independent of the choice of centered measure.

In addition, we prove an identification of HF₁ with LHF on polynomial growth groups for adapted, smooth, Abelian-centered measures. Next, for any finite-index subgroup H ≤ G and adapted smooth measure we shall see a quantitative induction-restriction principle: restriction along H and an explicit averaging operator give a linear isomorphism LHF(G, μ) ≅ LHF(H, μ_H), where μ_H is the hitting measure, with two-sided control of the Lipschitz seminorms. Separately, for quasi-isometries with bounded Abelian defect, we construct coarse harmonic coordinates that straighten them up to bounded error. This talk is based on joint work with Mayukh Mukherjee and Soumyadip Thandar.
Carl Schildkraut: Abelian structure in non-abelian approximate groups

Many problems in “additive” combinatorics are much easier in abelian groups. This difficulty can often be mitigated when the object in question has a relatively large abelian-like part. I will talk about a result of mine showing how to find these abelian-like parts in a few different settings and give a couple of applications.
Danielle West: The core of modules over free group algebras

Stallings core graphs are a famous and powerful graphical representation of finitely generated subgroups of a given free group. They enable geometric insights into the study of free groups, rendering many structural properties and algorithmic tasks intuitive. Consequently, they serve as a ubiquitous tool in combinatorial and geometric group theory. We introduce an analogue of this theory for modules over free group algebras. We define the core of a module and show that it is unique whenever the module is algebraic, also known as bound. Furthermore, we demonstrate how many of the core concepts from the theory of Stallings graphs, such as folding and pullbacks, generalize to these new objects. This framework enables us to introduce simple proofs and algorithms for many natural problems surrounding free group algebras.
Logan Richard: Lower semi-continuity of the fundamental group

Gromov–Hausdorff convergence formalizes convergence of sequences of metric spaces. A natural question to ask in this setting is when the topology of the limit may be related to the topology of the sequences which approximate it. One classical, positive answer to this question is “lower semi-continuity of the fundamental group”, which states that compact, locally contractible limits X of compact, geodesic spaces X_n satisfy that π₁(X) is a quotient of π₁(X_n), for n large enough. In particular, compact, locally contractible limits of compact, geodesic, simply connected spaces are themselves simply connected. We discuss progress toward proving analogues of this result in the non-compact setting.

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