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, TBA
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.
TBA