Seminars in Group Theory

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Ian Tan: Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

30th September, 2024

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(ℂ) × SL2(ℂ) → SO4(ℂ), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

Jonas Deré: Simply transitive NIL-affine actions of soluble Lie groups

26th August, 2024

Although not every 1-connected soluble Lie group G admits a simply transitive action via affine maps on ℝn, it is known that such an action exists if one replaces ℝn by a suitable nilpotent Lie group H, depending on G. However, not much is known about which pairs of Lie groups (G,H) admit such an action, where ideally you only need information about the Lie algebras corresponding to G and H. In recent work with Marcos Origlia, we show that every simply transitive action induces a post-Lie algebra structure on the corresponding Lie algebras. Moreover, if H has nilpotency class 2 we characterize the post-Lie algebra structures coming from such an action by giving a new definition of completeness, extending the known cases where G is nilpotent or H is abelian.

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.

Stuart White: Simple amenable C*-algebras

27th March, 2024

I'll give an overview of recent progress in the structure and classification of simple amenable C*-algebras, making parallels to the Connes-Haagerup classification of amenable von Neumann algebras and drawing examples from group actions.

Artem Chernikov: Recognizing groups in Erdős geometry and model theory

15th March, 2024

Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - variants of Hrushovski’s group configuration and of Zilber’s trichotomy principle - are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.

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).

Antonio Viruel: Permutation representations of finite groups via evolution algebras

18th December, 2023

In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X) ≅ G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.

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.

Ilya Skhredov: The Popularity Gap

4th December, 2023

Suppose that A is a finite, non-empty subset of a cyclic group of either infinite or prime order. We show that if the difference set A-A is 'not too large', then there is a non-zero group element with at least as many as (2+o(1))|A|2/|A-A| representations as a difference of two elements of A; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient 2 is best possible.

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.

Andrey Nikolaev: Non-standard polynomials and non-standard free group

20th October, 2023

Interpretation and bi-interpretation offer a novel approach to studying all structures elementarily equivalent to a given one. We use this approach to describe and study non-standard models of the ring of polynomials, Laurent polynomials, and the group ring of a free group.

In the presence of interpretation but not bi-interpretation, this approach produces a family of structures elementarily equivalent to a given one. We exploit this to introduce non-standard models of a free group. As time permits, we discuss their main properties.

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.

Nate Harman: Oligomorphic and Linearly Oligomorphic groups

11th September, 2023

Oligomorphic groups are a class of groups arising in model theory. I will discuss where these groups come from, highlight some of their interesting properties, and explain why I (a non-model-theorist) am interested in them. Then I will introduce a new notion of linearly oligomorphic groups, and give new examples of infinite-dimensional algebraic groups.

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.

Vladimir Vankov: Bestvina-Brady groups and generalizations

5th June, 2023

Right-angled Artin groups are perhaps the most ubiquitous manifestations of polyhedral products in geometric group theory and low-dimensional topology. The theory of their subgroups has been of great importance in the last couple of decades. This is especially true with regards to what are known as 'finiteness properties' - meaningful criteria for measuring ways in which infinite groups may behave like finite ones - as well as the theory of three-dimensional manifolds. We will visit some celebrated theorems and, if time allows, discuss problems arising from deck transformations of branched covering maps.