**Organiser**: Dominik Francoeur

This is a tentative list of upcoming speakers for the Algebra and Geometry Seminar at Newcastle University. Click on the name of the speakers to see the title and the abstract.

The schedule for the seminar can also be found in calendar form on this webpage.

**Time:** 14:00

**Title:** Spread and infinite groups

**Abstract:** The notion of the spread of a finite group was introduced by Brenner and Wiegold in 1975. A group G has non-zero spread if every non-trivial g in G is part of a generating pair. If we focus on groups which have non-zero spread, then this restricts our attention to 2-generated groups for which every proper quotient is cyclic. Surprisingly, this condition on quotients is sufficient for a finite group to have non-zero spread. This is a result of Burness-Guralnick-Harper from 2021 which has now appeared in Annals of Math. The world of 2-generated infinite groups is much more wild, e.g. there exist simple groups requiring k generators for any natural number k (it is currently open, however, whether all finitely presented simple groups are 2-generated). After covering some of the history, we will see that the result of Burness-Guralnick-Harper does not generalise to 2-generated infinite groups. We will do this by motivating a concrete example which acts naturally on the integers.

**Time:** 14:30

**Title:** Reflective centers of module categories

**Abstract:** In recent work with Chelsea Walton and Milen Yakimov we introduced the reflective center which is a braided module category associated to a given module category. In this talk, I will discuss this construction, which can be understood as an analogue of the Drinfeld center for module categories. In the case when the module category is given by modules of a comodule algebra A over a Hopf algebra H, the reflective center is given by modules of the reflective algebra of A. This reflective algebra is a crossed product of the original comodule algebra A and Majid’s covariantized Hopf algebra of H* (also known as the reflection equation algebra).

**Time:** 14:30

**Title:** Conjugacy geodesics and growth in dihedral Artin groups

**Abstract:** Like standard growth of (finitely generated) groups, one can define conjugacy growth of groups which, informally, counts the number of conjugacy classes in a ball of radius n in a Cayley graph. This was first studied by Riven for free groups, and techniques from geometry, combinatorics and formal language theory have proven to be useful for determining information about the conjugacy growth series for a variety of groups.

In this talk, I will first give an overview of what is know about the conjugacy growth series in groups, and some key tools in this area. Then we will apply these techniques to determine the conjugacy growth series for dihedral Artin groups, by considering conjugacy geodesics. This is joint work with my supervisor Laura Ciobanu.

**Time:** 14:30

**Title:** Subgroup separability and branch groups

**Abstract:** A group is said to be subgroup separable if all its finitely generated subgroups are closed in the profinite topology. This property can be seen as a stronger version of residual finiteness, and it has interesting algebraic and algorithmic consequences. Subgroup separability has been studied for various interesting classes of groups, including free groups, surface groups and fundamental groups of closed hyperbolic 3-manifolds. In this talk, we will be interested in studying subgroup separability of branch groups, a class of groups of automorphisms of rooted trees that plays an important role in the classification of just infinite groups. Part of this is joint work with Alejandra Garrido and Jone Uria-Albizuri.

**Time:** 14:00

**Title:** Growth of subsets

**Abstract:** The standard growth function of a finitely generated group counts the elements in the ball of radius n in the Cayley graph. We can modify this by only counting elements of some given subset of interest (for example a subgroup, or conjugacy class). We are concerned with both the asymptotics of this function, and the behaviour of the associated formal power series. In this talk I will discuss some work in progress on the growth of conjugacy classes in various classes of groups, before moving onto some more general results for subsets of virtually abelian groups. Joint work with Aram Dermenjian, and with Laura Ciobanu and Alex Levine.

**Time:** 14:00

**Title:** Markov chains on groups, and lumping by double cosets, with an application to the Random-to-top shuffle.

**Abstract:** This is joint work with Mark Wildon (Bristol). Let G be a group, and Q a probability measure on G. There is a Markov chain defined on G where transitions are given by right multiplication by an element of G randomly chosen according to Q. Recent work of Diaconis, Ram and Simper considers subgroups H and K of G such that the Markov walk driven by Q induces a Markov chain on the double coset space H\G/K; they give some conditions on Q for this to occur. We extend their analysis, stating a necessary and sufficient condition on Q, which takes a particularly nice form in the case that H=K. We look at an application of this theory to the case of a particular probability measure on the symmetric group Sn, connected to the random-to-top shuffle. Our approach reveals an unexpected connection with the theory of so-called involutory random walks, which allows us to calculate eigenvalues.

**Time:** 14:00

**Title:** ‘Odd’ complex Grassmannians are rigid

**Abstract:** The Grassmannian of \(m\)-planes in \(\mathbb{C}^{n+m}\) is a complex manifold that admits a canonical geometry known as the Fubini—Study metric. One of the main reasons to be interested this metric is that it is a solution to the (Riemannian) Einstein equations. About 40 years ago, Koiso discovered that there exist deformations of this metric that solve the Einstein equation to first order, raising the possibility that the Fubini—Study metric is actually part of a family of Einstein metrics. I will describe these deformations explicitly and report on work showing that for certain Grassmannians, one cannot deform the Fubini—Study metric to second order.

**Time:** 14:00

**Title:** Interpolation of open-closed TQFTs

**Abstract:** TQFTs are known to produce invariant of manifolds. In particular, an ordinary (oriented) closed 2d-TQFT with target the category of complex vector spaces will associate a complex number to each closed surface of genus g, giving thus a complex-valued sequence indexed by the natural numbers N. The sequences one could obtain that way are constrained by the nature of the target category. A work by Khovanov, Ostrik and Kononov explains how to interpolate between target categories in order to build TQFTs affording arbitrary sequences. I will show, using tools developed by my supervisor E. Meir, how to extend their work to the case of (oriented) open-closed 2d-TQFTs, and talk about the properties of these interpolating categories.

**Time:** 14:00

**Title:** Graph products: a taster

**Abstract:** Graph products of monoids provide a common generalisation of direct and free products. If the monoids in question are groups, then any graph product is a group. Graph products of finitely generated free monoids are also known as trace monoids, and arise in the study of concurrent processes in computer science. For groups there has been a largely separate development of both the theory and terminology. For example, a right-angled Artin monoid is a graph product of finitely generated free groups.

I will present some background to and history of these ideas, then move on to a discussion of questions asked in this area. In particular, how do certain algebraic and finitary conditions pass up and down between the constituent monoids and the graph product?

My results are joint with Nouf Alqahtani, Jung Won Cho, Yang Dandan and Nik Ruškuc.

**Time:** 14:30

**Title:** From strong contraction to hyperbolicity

**Abstract:** For almost 10 years, it has been known that if a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct a hyperbolic space where said element acts WPD. For a long time, the following question remained unanswered: if Morse is equivalent to strongly contracting, does there exist a space where all generalized loxodromics act WPD? In this talk, I will present a construction of a hyperbolic space, that answers this question positively.

**Time:** 14:30

**Title:** Left-orders on groups described by regular languages

**Abstract:** A group is left-orderable if it admits a total order that it is invariant under the left-multiplication action.
Left-orders are a useful extra structure on groups. In this talk we will focus on describing left-orders with a finite state automaton that codifies which elements are greater than the identity. We will give examples of such orders and describe some groups that do not admit such orders. Based on works with Juan Alonso, Joaquin Brun, Hang Lu Su and Cristobal Rivas.

**Time:** 14:30

**Title:** Decomposing tensor powers

**Abstract:** I will explain how to decompose tensor powers \(V^{\otimes r}\) of the natural representation \(V = \mathbb{C}^{m|n}\) of the orthosymplectic supergroup \(OSp(m|2n)\) into indecomposable summands. Along the way we will meet Deligne's interpolating category \(Rep(O_t)\), \(t \in \mathbb{C}\), and modules of Khovanov's arc algebra of type B. The first half of the talk will be a general introduction to the problem and will be more like a colloquium talk than a seminar talk.

**Time:** 14:30

**Title:** Commutative triples and matrix orthogonal polynomials

**Abstract:** There is strong relation between Gelfand pairs and special functions. Commutative triples are extensions of Gelfand pairs. We discuss a general framework on how to obtain a connection to matrix orthogonal polynomials in case we have a compact commutative triple with some additional requirements. Some examples arising from symmetric pairs are discussed.

**Time:** 14:30

**Title:** Full runner removal theorem for Ariki-Koike algebras

**Abstract:** The determination of the decomposition numbers, i.e., the composition multiplicities of the simple modules D in the Specht modules S is one of the most important outstanding problems in representation theory of the symmetric groups and related algebras. For the Iwahori–Hecke algebras of the symmetric group Fayers proved a theorem which relates decomposition numbers for different values of e, by adding ‘full’ runners to the abacus displays for the labelling partitions. I will describe the background of Fayers’ theorem, and then talk about a ‘runner removal’ theorem for Ariki–Koike algebras that generalises Fayers’ one.

**Time:** 14:30

**Title:** Subgroups of even Artin groups of FC type

**Abstract:** Even FC-type Artin groups exhibit a finite index subgroup satisfying the strongest Tits alternative: every subgroup is either finitely generated abelian, or it maps onto a non-abelian free group. Parabolic subgroups play a fundamental role, especially their closure properties under intersections and taking roots. This is a joint work with Prof. Yago Antolín.

**Time:** 14:30

**Title:** Some aspects of Khovanov homology

**Abstract:** This mostly expository talk introduces Khovanov homology in the context of low-dimensional topology. Some of its most important applications shall be discussed, with an emphasis on geometric results. Time permitting, some current work in progress shall be presented.

**Time:** 14:30

**Title:** Topological Full Groups and Stein's Groups

**Abstract:** Topological full groups are a way to build interesting examples of groups using dynamical systems, in particular, examples of infinite simple groups with various finiteness properties. In this talk, I will introduce topological full groups and explain how I have used the framework of topological full groups to understand a class of groups of piecewise linear bijections that generalise Thompson's group \(V\), which were introduced by Melanie Stein.

**Time:** 14:30

**Title:** Towards ''super cluster algebras'' of type A

**Abstract:** In the study of cluster algebras, computing the generators (cluster variables) explicitly is an important problem. For surface cluster algebras, one can do this combinatorially, using dimer covers of snake graphs. Recent work by Musiker, Ovenhouse and Zhang extend this approach to ''super cluster algebras'' of type A. Alternatively, in the classic surface cluster algebras setting, one can use a representation theoretic approach to compute cluster variables explicitly using the CC-map. Motivated by this, we introduce a representation theoretic interpretation and a super CC-map in the super algebras setting. This is a joint work in progress project with Canakci, Garcia Elsener and Serhiyenko.