Seminars in Infinite Groups

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Susan Hermiller: Subgroups of the group of dyadic piecewise linear homeomorphisms of the real line

3rd May, 2024

The group of dyadic orientation-preserving piecewise linear (PL) homeomorphisms of the unit interval is called Thompson's group F, and the question of which groups are - or cannot be - subgroups of F has yielded many interesting results. In this talk I'll discuss the question of what groups can or cannot be subgroups of Aut(F) (the automorphism group of F), and more particularly subgroups of an index 2 subgroup of Aut(F) that is isomorphic to a group of dyadic PL homeomorphisms of the real line.

Laura Ciobanu: Group equations, constraints and decidability

8th March, 2024

In this talk I will discuss group equations with non-rational constraints, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications. I will introduce equations in groups and several kinds of constraints, and show that equations with length, abelian or context-free constraints are decidable in virtually abelian groups (joint with Alex Evetts and Alex Levine). This contrasts the fact that solving equations with abelian constraints is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation (joint work with Albert Garreta).

Markus Lohrey: Streaming word problems

8th December, 2023

We are interested in highly efficient algorithms for word problems of groups: the algorithm should read the input word once from left to right symbol by symbol (such algorithms are known as streaming algorithms), spending ideally only constant time for each input letter. Moreover, the space used by the algorithm should be small, e.g. O(log n) if n is the length of the input word. To achieve these goals we need randomization: the algorithm is allowed to make random guesses and at the end it gives a correct answer (is the input word trivial in the underlying group?) with high probability. We show that for a large class of groups such algorithms exist, where in particular the space complexity is bounded by O(log n). These groups are obtained by starting with finitely generated linear groups and closing up under the following operations: finite extensions, graph products, and wreath products where the left factor is f.g. abelian. We also contrast this result with lower bounds. For instance, for Thompson’s group F every randomized streaming algorithm for the word problem for F has space complexity Ω(n) (n is again the length of the input word).

Enric Ventura: The central tree property and some average case complexity results for algorithmic problems in free groups

8th December, 2023

We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude lower than the worst case complexity of the best known algorithms. This applies both to subgroups given by a fixed number of generators, and to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. Another application is given to the average case complexity of the relative primitivity problem, using Shpilrain's recent algorithm to decide primitivity in a free group, whose average case complexity is a constant depending only on the rank of the ambient free group.

Scott Balchin: A jaunt through the tensor-triangular geometry of rational G-spectra for G profinite or compact Lie

26th October, 2023

In this talk, I will report on joint work with Barnes-Barthel and Barthel-Greenlees which analyses the category of rational G-equivariant spectra for G a profinite group or compact Lie group respectively. In particular, I will focus on a series of results regarding the Balmer spectra of these categories, and how the topology of these topological spaces informs structural results regarding the category.

Ilya Kapovich: Primitivity index bounds in free groups, and the second Chebyshev function

13th October, 2023

Motivated by results about 'untangling' closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, dprim(g;FN) and dsimp(g;FN), where 1 ≠ g ∈ FN, and obtained some upper and lower bounds for these functions. In this talk, we study the behaviour of the sequence dprim(anbn; F(a,b)) as n → ∞. Answering a question of Kapovich, we prove that this sequence is unbounded and that for ni=lcm(1,2,...,i), we have |dprim(anibni; F(a,b))-log(ni)| = o(log(ni)). By contrast, we show that for all n ≥ 2, one has dsimp(anbn;F(a,b)) = 2.

In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.

Alexei Miasnikov: Musings on exponentiation in groups

29th September, 2023

Exponentiation in groups is an old and well-researched subject. The main theme here is to understand what a 'non-commutative module' is in various classes of groups. Following Lyndon in 1994 V. Remeslennikov and myself introduced a notion of a group admitting exponentiation in an associative unitary ring R (now called R-groups). This is the most 'freest and universal' exponentiation that works in all groups and it applies nicely to free and hyperbolic groups, free products with amalgamation and HNN extensions, etc. M. Amaglobeli started studying R-groups in varieties, in particular, nilpotent and solvable ones. However, if a group satisfies an identity the notion of exponentiation can be further adjusted to reflect more closely the nature of the group. Thus, in the class of nilpotent groups there is a famous P. Hall and A. Mal'cev's exponentiation that gives a perfect notion of a 'nilpotent non-commutative module'. Recently, working on first-order properties of free metabelian groups, we together with O. Kharlampovich explored an exponentiation that naturally occurs in metabelian groups. In this talk I will discuss all these exponentiations, the corresponding centroids and tensor completions, and how they relate to each other.

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Michael Wibmer: Expansive endomorphisms of profinite groups

25th May, 2023

Étale algebraic groups over a field k are equivalent to finite groups with a continuous action of the absolute Galois group of k. The difference version of this well-known result asserts that étale difference algebraic groups over a difference field k (i.e., a field equipped with an endomorphism) are equivalent to profinite groups equipped with an expansive endomorphism and a certain compatible difference Galois action. In any case, understanding the structure of expansive endomorphisms of profinite groups seems a worthwhile endeavour and that's what this talk is about.

Emma Brink: Condensed Group Cohomology

18th April, 2023

The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of G, M (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of π0(G) with coefficients in M.

Generalizing results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological δ-functor in the condensed setting for a large class of topological groups.

Martina Conte: Definability of the rank and the dimension of p-adic analytic pro-p groups

31st March, 2023

Recently, Nies, Segal and Tent started an investigation of finite axiomatizability in the realm of profinite groups. Among the classes of profinite groups under their consideration is the class of p-adic analytic pro-p groups. In joint work with Benjamin Klopsch, we consider two key invariants of these groups, namely rank and dimension, and show that they can be characterized by a single first-order sentence. Before discussing these results I will introduce the relevant background. If time permits, I will also present some natural generalisations.

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.

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.

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.

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.