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.
Let G be a primitive permutation group on a finite set X and recall that a subset of X is a base for G if its pointwise stabiliser is trivial. The base size of G, denoted b(G), is defined to be the minimal size of a base. This natural invariant has been intensively studied for many years, finding a wide range of applications. In this talk I will report on recent progress concerning a project initiated by Jan Saxl in the 1990s, which seeks to determine all the primitive groups with b(G) = 2. I will also outline some of the main applications and I will highlight one or two related problems.
By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.
A set of permutations A is said to contain a product if there are two permutations in it whose product also lies in A; otherwise A is called product free. We show that the density of a product free set A ⊆ An is at most O(1/√n). This result is asymptotically tight, and improves on earlier results of Gowers (O(1/n1/3)) and Eberhard (O(log7/2 n /√n)). We also show stability results. Our proof uses recent refinements of the hypercontractive inequality over Sn for the class of "global functions", and in particular a version of the level 1 inequality in this setting. Based on a joint work with Peter Keevash and Noam Lifshitz.
The concept of a synchronizing permutation group was introduced nearly 15 years ago as a possible way of approaching The Černý Conjecture. Such groups must be primitive. In an attempt to understand synchronizing groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, ℚI-groups, spreading, separating, synchronizing, almost synchronizing and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where.
A graph is vertex-transitive if its group of automorphism acts transitively on its vertices. A very important concept in the study of these graphs is that of local action, that is, the permutation group induced by a vertex-stabilizer on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalize it to the case of directed graphs.
A non-compact, compactly generated, locally compact group whose proper quotients are all compact is called just-non-compact. Discrete just-non-compact groups are John Wilson’s famous just-infinite groups. In this talk, I'll describe an ongoing project to use permutation groups to better understand the class of just-non-compact groups that are totally disconnected. An important step for this project has recently been completed: there is now a structure theorem for non-compact tdlc groups G that have a compact open subgroup that is maximal. Using this structure theorem, together with Cheryl Praeger and Csaba Schneider’s recent work on homogeneous cartesian decompositions, one can deduce a neat test for whether the monolith of such a group G is a one-ended group in the class 𝒮 of non-discrete, topologically simple, compactly generated, tdlc groups. This class 𝒮 plays a fundamental role in the structure theory of compactly generated tdlc groups, and few types of groups in 𝒮 are known.
