Heiko Dietrich: Canonical presentations for finite solvable groups
Abstract: We define and describe how to compute canonical group presentations for finite solvable groups, such that two finite solvable groups are isomorphic if and only if they have the same canonical presentation. Our approach is motivated by O'Brien's (1993) canonical presentations for groups of prime power order and utilises ideas from group cohomology first described by Robinson (1982), Smith (1994), and Holt (2001). This is part of a larger project concerned with group identification and construction-by-ID.
Florian Lehner: Graphs of group actions and groups acting on trees
Abstract: Bass-Serre theory is perhaps the most important tool for analysing group actions on trees. It relates group actions on trees to graphs of groups, allowing a description of the groups as iterated amalgamated free products and HNN extensions. Conversely, given a graph of groups, it allows us to construct a group action on a tree. In practice, however, obtaining new examples of groups acting on trees through this construction can be difficult, as vertex and edge groups must satisfy rigid compatibility conditions. In a foundational paper motivated by a question of Serre, Tits initiated an alternative approach focused on permutational properties of group actions on trees. The ideas in Tits’ paper - in particular property (P), which enforces independence of actions on disjoint subtrees - have been highly influential. For instance, they underpin the universal groups of Burger and Mozes as well as Smith’s box product construction, which were later combined by Reid and Smith into the theory of local action diagrams. The aim of this talk is to introduce graphs of group actions, a notion simultaneously generalising graphs of groups and local action diagrams. The resulting theory retains the full generality of Bass-Serre theory while preserving the flexible local constructability of local action diagrams, allowing for instance the systematic construction of P_k-closed groups. Based on joint work with Matthias Chan, Christian Lindorfer, Rögnvaldur Möller, and Wolfgang Woess.
Cheryl Praeger: Primitive actions in, and invariable generation of, alternating groups by finite simple subgroups
Abstract: The talk will report on several recent interlinked investigations about primitive actions of finite simple groups and the issue of invariable generation by them. Regarding primitive actions, it turns out that, with the single exception of the simple group PSL(2,11), every finite nonabelian simple group X occurs as a primitive and almost maximal subgroup of some alternating group Alt(n), that is to say, X is primitive and Aut(X)\cap Alt(n) is maximal. Our question regarding invariable generation was: given any finite simple groups X and Y, is it always possible to find an integer n and embeddings of X, Y as subgroups of Alt(n) such that, replacing X and Y by any arbitrary conjugates always yields two simple subgroups that generate Alt(n), that is to say, the embedded subgroups X and Y invariably generate Alt(n). We believe that the answer is “just about always yes” – and we will surely have the answer by November! This is joint work with Luke Morgan and Alexandre Zalesski
Kamilla Rekvenyi: The orbital diameter of primitive permutation groups
Abstract: Let G be a group acting transitively on a finite set Ω. Then G acts on Ω ×Ω componentwise. Define the orbitals to be the orbits of G on Ω ×Ω. The diagonal orbital is the orbital of the form ∆= {(α, α)|α∈Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set Ω and edge set (α, β) ∈Γ with α, β∈Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs. There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding explicit bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. Finally, I will discuss some results for almost simple groups, which is joint work with Attila Maróti.
Anne Thomas: Divergence and hypergraph index for Coxeter groups
Abstract: The divergence of a pair of geodesic rays measures how fast they move away from each other. In the 1990s, Gersten used this idea to define a quasi-isometry invariant for finitely generated groups, also called divergence, and divergence has since been investigated for many families of groups of importance in geometric group theory. In this talk, we discuss progress on understanding divergence in (infinite) Coxeter groups. The right-angled case is now well-understood, and we have a partly conjectural picture for general Coxeter groups. This involves a combinatorial invariant for Coxeter systems called hypergraph index, which was introduced in the right-angled case by Levcovitz. This includes joint work and work-in-progress with Pallavi Dani, Max Mikkelsen, Yusra Naqvi and Ignat Soroko.
George Willis: Self-replication and scale groups
Abstract: Self-replicating and scale groups are two classes if groups acting on infinite trees that arise in quite different contexts but are essentially equivalent. These groups may be roughly classified by their local action, which is a transitive finite permutation group. I will explain the backgrounds of these classes of groups and the relationship between them before going on to describe joint work with Stephan Tornier that seeks to make a finer classification by investigating subgroups of iterated wreath products of the local action.
Vishnuram Arumugam: Multicirculants and Generalised Quadrangles
Abstract: Multicirculant structures on the incidence and point graphs of finite generalized quadrangles of order $(q,q)$, where $q>1$, are considered. It is proven that if there exists a semiregular automorphism of the incidence graph that interchange points and lines with $m$ vertex-orbits and $2q$ is not a square, then $m=a(q^2+1)$ for some divisor $a$ of $q+1$. For the point graph, it is shown that it is neither a circulant nor a bicirculant, while it is a tricirculant if and only if $q=2$. For the incidence graph it is shown that it is neither a circulant nor a bicirculant, it is a tricirculant if and only if $q=2$, it is never a tetracirculant, can be a pentacirculant only if $2q$ is a square, and it is a hexacirculant if and only if $q=2$.
Gustav Bagger: Generators of the unit group of finite fields
Abstract: A primitive element of a finite field F is a generator of its unit group F*. Finding such elements is entirely non-trivial and this search has given rise to many interesting results in number theory. If one is willing to assume the generalised Riemann hypothesis, then every (non-silly) integer is primitive to infinitely many prime order fields. Moreover, the least primitive element in such fields is less than the square root of the order. I will present an effective method for finding primitive elements in finite fields with prime-power order. This method is an amalgamation of my hybrid bound on prime divisors and joint work with J. Punch on prime sieves.
Luka Carroll: The epimorphism problem onto the bicyclic monoid
Abstract: The bicyclic monoid B is a fundamental algebraic object consisting of words over the alphabet {a,b} subject only to the relation ab=e where e is the empty word. It plays an important role in the study of formal languages (particularly Dyck languages) and in semigroup theory. This talk will discuss recent work analysing the complexity of the following decision problem: on input a finitely presented inverse monoid M, decide if there exists a surjective homomorphism from M onto B.
Lei Chen: Covering a group by the conjugates of a cost
Abstract: It is well known that for a group G and a subgroup A of G it is impossible to cover G with the conjugates of A. Thus, instead of the conjugate of A we take the conjugate of the coset Ax in G and check if the union of (Ax)^g covers G-{1} for g in G. Moreover, if (Ax)^g covers G for all Ax in Cos(G:A), we say that (G,A) is CCI. We are aiming to classify all such pairs. It has been proven by Baumeister-Kaplan-Levy that this can be reduced down to the case where A is maximal in G and so that the action of G on Cos(G:A) is primitive. and they showed that (G,A) is CCI if G is 2-transitive. By O'Nan-Scott Theorem and CFSG, we see that G is either an affine group or almost simple. In the paper of Baumeister-Kaplan-Levy, it has been shown that affine CCI groups are 2-transitive. Thus, it remains to consider the almost simple groups. By employing the knowledge of buildings, representation theory and Aschbacher-Dynkin theorem, we prove that, apart from finitely many small cases, the CCI almost simple groups are 2-transitive.
Marcus Chijoff: A GAP Package For Local Action Diagrams
Abstract: Recent work by Reid-Smith parameterised closed subgroups acting on trees satisfying J. Tits' Property (P). This is done in terms of relatively simple combinatorial structures called Local Action Diagrams. Properties of the group acting on a tree can be determined from the corresponding local action diagram and so when a local action diagram is finite it is natural to compute these properties using a computer program. This talk serves as both an introduction to local action diagrams and also shows how a work-in-progress GAP package implementing them can be used to calculate various properties of a given finite local action diagram (and so the associated group).
Zhaochen Ding: On the Cartesian products of graphs that are Cayley graphs
Abstract: I will discuss Cartesian product and regular subgroup of wreath product.
Fabricio Dos Santos: Weak order on groups generated by involutions
Abstract: An involution system (W,S), that is a group W generated by a set of involutions S, is naturally endowed with a weak order arising from orienting its Cayley graph. If (W,S) is a Coxeter system, Björner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their related structures. In this talk, we plan to discuss the following question: For which involution systems is the weak order a complete meet-semilattice? The class of involution systems that satisfies this condition is larger than the class of Coxeter systems (it contains, for instance, Cactus groups). In the case of an involution system with sign character, we provide a finite presentation by generators and relations and a classification in rank 3. If time allows, we will also discuss open problems (e.g. in relation to automatic structures, geometric representations, CAT(0) cell complexes, ...). Joint work with Christophe Hohlweg and Aleksandr Trufanov.
Dilpreet Kaur: Essential Dimension of small finite groups
Abstract: We discuss the Essential Dimension of finite groups and its connection with representation dimension. We use this connection to compute the essential dimension of some small finite groups.
Melissa Lee: Elusive groups and the Polycirculant Conjecture
Abstract: A derangement is a permutation with no fixed points. A classical theorem of Jordan asserts that every primitive group contains a derangement. However, there exist transitive groups with no derangements of prime order -- such groups are called elusive. Elusive groups have been widely studied over the past several decades, partly due to their connection to Polycirculant Conjecture and Semiregularity Problem. In this talk I will discuss very recent joint work with Saul Freedman, settling both of these problems, which had been open for 40 years.
Shir Levav Porat: Specialising Belyi maps
Abstract: This talk is about the effective inverse Galois problem. Given a finite group G, the task is to produce an integral polynomial whose Galois group is G. Famously, the problem was recently resolved for the sporadic simple group M23 using Belyi maps. With Zachary Couvillion and John Voight, we give results about specialising Belyi maps which resolve the inverse Galois problem in new and old cases.
Euan Mendoza: Strong Krull-Remak-Schmidt type theorem for Mixed Tensors with Algorithmic Applications
Abstract: Mixed tensor isomorphism captures several related isomorphism notions such as p-groups of class 2 and exponent p isomorphism, homogenous degree d polynomial isomorphism and (Lie) algebra isomorphism. Individually there has been several Krull-Remak-Schmidt type decomposition results for each of these objects, such as Wilson (2010), Harrison (1977), Strassen (1987). Our contribution is first to present a unifying framework for these results with a stronger condition then the usual Krull-Remak-Schmidt type theorem, that each decomposition is unique into embedded subtensors. A key ingredient for such a result is the notion of a centroid algebra for mixed tensors. Additionally we show that such decompositions can be computed efficiently, and present some applications to theoretical computer science such as reducing the so-called mixed tensor isomorphism problem to the computational mixed tensor automorphism problem. This is joint work with Murray Elder, Chang (Aaron) He, Luke Mathieson and Youming Qiao.
Luke Morgan: Almost maximal primitive groups
Abstract: The O'Nan-Scott Theorem divides (finite) primitive permutation groups into several types. Among the most important classes of these are the almost simple groups. For a given n, determining the almost simple groups that arise as maximal subgroups of Alt(n) is a long-studied and difficult question. Conversely, for a fixed simple group T, one can ask whether there exists an integer n for which (the normaliser of) T arises as a maximal subgroup of Alt(n). Remarkably, there is only one finite simple group for which the answer is no.
Jeff Saunders: Oriented transformation geometry
Abstract: Transformation geometry, reductively, is the study of Euclidean geometry through the conjugation action of the Euclidean group E(2) on itself: lines are identified with reflections in themselves, and points with half-turns about themselves. In this talk I use transformation geometry to present a tempting but flawed proof of a nice property of odd polygons. Working in a double cover of E(2) adds orientations to lines and points, and allows the proof to be repaired. This double cover lives in a particular Clifford algebra, leading us to so-called geometric algebra.
Colin Reid: Uncountably many local isomorphism types of compactly generated simple groups
Abstract: Let $\mathscr{S}$ be the class of non-discrete totally disconnected locally compact groups that are compactly generated and topologically simple. We have some general theory for groups in $\mathscr{S}$ but are a long way from any classification. By analogy with Lie theory, you might try some approach based on understanding what $G \in \mathscr{S}$ looks like close to the identity, and then extending to a global description. A basic question is: how many possibilities are there at each stage?
In 2017, Simon Smith gave a construction that produces $2^{\aleph_0}$ different groups in $\mathscr{S}$ that are locally isomorphic to one another. We complement this result by producing $2^{\aleph_0}$ groups in $\mathscr{S}$ that are \emph{not} locally isomorphic to one another. I will explain the construction, which builds on Smith's construction and a 2016 construction by Ilir Snopce of $2^{\aleph_0}$ metabelian uniform pro-$p$ groups.
Joint work with: I. Castellano, J. Fariña-Asategui, M. Eguzki Garciarena, B. Marchionna, M. Quick, S.M. Smith, S. Tornier, M. Vannacci, J.S. Wilson. This collaboration started with a problem session at the June 2026 workshop "Branch groups: subgroups, rigidity, topologies" (organizers: D. Francoeur, A. Garrido and T. Nagnibeda) at the Universidad Complutense de Madrid.
Gabriel Verret: Orders of quotients of groups and applications to locally-transitive graphs
Abstract: We will discuss a result classifying the possible growth types of the set of orders of finite quotients of a finitely generated group. We also discuss applications of this result to the set of orders of various families of symmetric graphs.
Bailey Whitbread: A Deligne-Simpson problem
Abstract: A problem in group theory dating back to Frobenius goes as follows. Given some conjugacy classes in a group, do there exists elements in each class whose product equals 1. Choosing the group of invertible complex matrices gives rise to the Deligne–Simpson problem. This talk is about a younger geometric version of that problem. With Masoud Kamgarpour, we solve this problem in all open cases by exploiting a connection to the world of braids.