Schedule for: 16w5087 - Permutation Groups

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is concerned with locally compact groups admitting a continuous 2-transitive action on a compact space. Two fundamental classes of such are exhaustively understood: the finite 2-transitive groups and the 2-transitive Lie groups. The groups defined in the title form a third such class on which we shall focus. A basic feature is that all groups in that class have a simple socle. A key challenge is to determine which simple groups occur in this way. We will survey characterizations and classification results obtained recently in that direction.

A graph is vertex-primitive if its automorphism group does not preserve any nontrivial partition of its vertex-set. It is an easy exercise to prove that (apart from some trivial exceptions) a vertex-primitive graph cannot have distinct vertices with equal neighbourhoods. I will discuss some results about vertex-primitive graphs having two vertices with “almost” equal neighbourhoods, and how these results were used to answer a question of Araújo and Cameron about synchronising permutation groups. These results were also the motivation for a recent classification of vertex-primitive graphs of valency 5. (Graphs of valency at most 4 had previously been classified.) I will describe this classification, some of the issues that arose in the proof, and the connection with the previous problem.

I'll discuss a theorem showing that no large member of a certain class of lattices is an overgroup lattice in an exceptional group of Lie type. The proof involves a question about pairs $(M_1,M_2)$ of maximal subgroups of a finite group $G$ such that the overgroup lattice in $G$ of $M_1\cap M_2$ is a 2-simplex.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A partial linear space consists of a non-empty set of points P and a collection of subsets of P called lines such that each pair of points lies on at most one line, and each line contains at least two points. A partial linear space is proper if it is not a linear space or a graph. In this talk, we will consider some recent progress on classifying the finite proper partial linear spaces with a primitive affine automorphism group of rank 3.

I will give a brief overview of (as far as I know) the current state of the putative classification of all Lie-primitive subgroups of the exceptional algebraic groups, and its implications for the maximal subgroup structure of the finite (almost simple) exceptional groups of Lie type.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Intransitive and imprimitive permutation groups preserve disjoint union decompositions and are routinely studied by considering their actions on invariant partitions. I would like to present a similar approach to the study of permutation groups that preserve cartesian product decompositions. Such groups occur naturally in the various versions of the O'Nan-Scott Theorem, and also in combinatorial applications, such as groups of automorphisms of Hamming graphs. Much of the theory I present is valid for arbitrary permutation groups. However, combining this theory with the classification of finite simple groups leads to a surprisingly detailed descriptions of finite groups that act on cartesian products. The results I present were obtained in collaboration with Robert Baddeley and Cheryl Praeger.

We first recall a completion operation which takes as input a group with a commensurated subgroup and outputs a locally compact group. This operation allows one to study finitely generated groups via locally compact groups and vice versa. We apply this completion to study the compelling class of finitely generated branch groups. In particular, we show every commensurated subgroup of a just infinite finitely generated branch group is either finite or of finite index.

If $(K,f)$ is a difference field, and a is a finite tuple in some difference field extending $K$, and such that $f(a)$ in $K(a)^{alg}$, then we define $dd(a/K) = lim [K(f^k(a),a):K(a)]^{1/k}$, the distant degree of $a$ over $K$. This is an invariant of the difference field extension $K(a)^{alg}/K$. We show that there is some $b$ in the difference field generated by $a$ over $K$, which is equi-algebraic with $a$ over $K$, and such that $dd(a/K)=[K(f(b),b):K(b)]$, i.e.: for every $k>0$, $f(b)$ in $K(b,f^k(b))$. Viewing $Aut(K(a)^{alg}/K)$ as a locally compact group, this result is connected to results of Willis on scales of automorphisms of locally compact totally disconnected groups. I will explicit the correspondence between the two sets of results.

In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this talk, we will discuss the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables. This is joint work with Van H. Vu.

The subgroup structure of reductive groups has been intensively studied since at least the 1950s, when Dynkin classified maximal connected subgroups of complex reductive groups. One of the main outstanding problems in the theory is to classify so-called "non-completely-reducible" reductive subgroups. Recent joint work with Adam Thomas has achieved such a classification for subgroups of exceptional simple groups, when the characteristic is 'good'. When the characteristic is bad, the theory becomes much more delicate, and we will discuss some of the problems arising, including understanding 'non-abelian cohomology sets' which arise.

Given a totally disconnected, locally compact group $G$, M\”oller gave a graph-theoretic characterisation for subgroups tidy for $x\in G$. We consider a corresponding result for flat subgroups of $G$.

A transitive permutation group is called semiprimitive if each normal subgroup is transitive or semiregular. This large class of groups includes the classes of primitive, quasiprimitive, innately transitive and Frobenius groups. Apart from being a generalisation of these important classes of permutation groups, motivation to study this class came from problems in abstract algebra and in algebraic graph theory. A barrier to their study has been the lack of any apparent structure and the prevalence of wild examples. In this talk I will report on joint work with Michael Giudici in which we brought some clarity to the study of this class of groups. We found that there is strong structure to a semiprimitive group, although not as precise as the O'Nan-Scott Theorem for primitive groups and there is rough structure that explains how semiprimitive groups are built from innately transitive groups. Along the way I'll mention plenty of examples and time permitting some application of this theory to the motivating problems.

It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years, such as random generation, (2,3)-generation and so on. In this talk I will report on recent joint work with Martin Liebeck and Aner Shalev on similar problems for subgroups of (almost) simple groups, focussing on maximal and second maximal subgroups. We prove that every maximal subgroup of a simple group can be generated by four elements (this is best possible) and we show that the problem of determining a bound on the number of generators for second maximal subgroups depends on a formidable open problem in Number Theory. Time permitting, we will also present some related results on random generation and subgroup growth.

The full symmetric group S on a countably infinite set X carries a natural topology, the topology of pointwise convergence. I will discuss joint work in progress with Cheryl Praeger and Simon Smith, and some open questions, on multiply transitive locally compact subgroups of S, on maximal-closed subgroups of S, and on subgroups of S which are maximal subject to being subdegree-finite.

For a topological group G several cohomology theories have been introduced
and studied in the past. In many cases the main motivation was to obtain an
interpretation of the low-dimensional cohomology groups in analogy to discrete
groups. The aim of this talk is to give firstly interpretations of the first degree rational discrete cohomology functor $\mathrm{dH}^1(G,-)$ introduced in [3], where $G$ is a totally disconnected locally compact (=t.d.l.c.) group. Secondly, it will be shown how these interpretations can be used to prove several results about t.d.l.c. groups in analogy with the discrete case. Namely, we prove that a non-trivial splitting of a compactly generated t.d.l.c. group can be detected by knowing a single cohomology group in analogy to [1, Theorem IV 6.10]. As a consequence, we characterize a compactly presented t.d.l.c. group of rational discrete cohomological dimension 1 to be a fundamental group of a finite graph of profinite groups in analogy to [2, Theorem 1.1] 1. Dicks, W. and Martin John Dunwoody. Groups acting on graphs. Vol. 17. Cambridge University Press, 1989.
 2. Dunwoody, M. J. Accessibility and groups of cohomological dimension one. Proceedings of the London Mathematical Society 3.2 (1979): 193-215.
 3. Castellano, I., and Th Weigel. Rational discrete cohomology for totally disconnected locally compact groups. Journal of Algebra 453 (2016): 101-159.

An $s$-arc in a digraph $\Gamma$ is a sequence $v_0,v_1,\ldots,v_s$ of vertices such that for each $i$ the pair $(v_i,v_{i+1})$ is an arc of $\Gamma$. There are several important differences between the study of $s$-arc-transitive graphs and $s$-arc transitive digraphs. For example, there are no 8-arc-transitive graphs of valency at least 8, while for every positive integer $s$ there are infinitely many digraphs of valency at least three that are $s$-arc-transitive but not $(s+1)$-arc transitive. In this talk I will discuss a solution to an old question of Cheryl Praeger about the existence of vertex-primitive 2-arc-transitive digraphs. This is joint work with Cai Heng Li and Binzhou Xia.

This talk is about infinite permutation groups that satisfy a finiteness condition: all orbits of point stabilizers are finite. Such groups are called ${\em subdegree-finite}$. Subdegree-finite permutation groups are the natural permutation representations of totally disconnected and locally compact topological groups.

In this talk I'll present a recent result which describes the structure of all subdegree-finite primitive permutation groups. It is akin to the seminal O'Nan--Scott Theorem for finite primitive permutation groups.

Studying the primitive actions of a group corresponds to studying its maximal subgroups. In the case where the group is countably infinite, one of the first questions one can ask is whether there are any primitive actions on infinite sets; that is, whether there are any maximal subgroups of infinite index. The study of maximal subgroups of countably infinite groups has so far mainly concerned classes of groups where the Tits Alternative holds (every subgroup is either virtually soluble or contains a free subgroup) and in the case where there are maximal subgroups of infinite index, there are uncountably many. It is natural to investigate this question for groups of intermediate growth, for instance, some groups of automorphisms of rooted trees. I will report on some recent joint work with Dominik Francoeur where we show that some groups of intermediate growth have exactly countably many maximal subgroups of infinite index.

Idempotents ($e^2=e$) in (nonassociative commutative) algebras (e.g., Jordan algebras) behave sometimes like involutions in a group. Indeed, in some (very) interesting cases one can associate an involutive automorphism of the algebra to an idempotent. This topic is related to Griess algebras (and the Monster group), Majorana algebras, Axial algebras (and the Fischer groups) and Miyamoto involutions. We explore mostly algebras generated by two idempotents, having in mind groups, and an extension of the results to algebras generated by finitely many idempotents (when possible). Joint work with Louis Rowen.

In joint work with Itay Kaplan, we show that $AGL_n(Q)$ ($n>1$) and $PGL_n(Q)$ ($n>2$) are maximal amongst closed proper subgroups of the infinite symmetric group. I will present this result and its proof which relies on Adeleke and Macpherson's classification of infinite Jordan groups. I will also mention some open questions.

I will be talking about joint work with Phillip Wesolek. A chief factor of a topological group $G$ is a factor $K/L$, where $K$ and $L$ are closed normal subgroups such that no closed normal subgroup of $G$ lies strictly between $K$ and $L$. We show that a compactly generated locally compact group admits an 'essentially chief series', that is, a finite normal series in which each of the factors is compact, discrete or a chief factor. In the totally disconnected case, the proof is based on the fact that $G$ acts vertex-transitively on a locally finite connected graph with compact open stabilizers. I will also indicate why totally disconnected chief factors can have a complicated normal subgroup structure as groups in their own right, in contrast to semisimple groups.

The closed subgroups of the group of permutations of N coincide with the automorphism groups of countable structures. We consider classes of such groups, such as being oligomorphic, or being topologically finitely generated. We study their complexity in the sense of descriptive set theory. If the class is Borel, we next consider the complexity of the topological isomorphism relation. For instance, the classes of locally compact and of oligomorphic groups are both Borel. In either case, the isomorphism relation is bounded in complexity by the problem of deciding whether two countable graphs are isomorphic. In the first case, and even restricted to compact (i.e. profinite separable) groups, this upper bound is known to be sharp. The upper bounds have been obtained independently by Rosendal and Zielinski (arXiv, Oct. 2016). This is joint work with A. Kechris and K. Tent.

In this talk we discuss results about generation and presentations of Kac-Moody groups over finite fields and their consequences for some Chevalley groups. This talk is partly based on recent work with A. Lubotzky and B.Remy, and with D. Rumynin.

We propose a conjectural upper bound for the number of irreducible Brauer characters in a $p$-block of a finite group, argue that it holds for $p$-solvable groups, give some further evidence in the case of quasi-simple groups and discuss some reductions. This is joint work with G. Robinson.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.