Groups 2012: Abstracts

(as of Jan 8)

Dave Benson: Cohomology of groups: a crossroads in mathematics.

This talk is designed as an introduction to the cohomology of groups, and particularly of finite groups. I shall describe the algebraic, topological and number theoretic views of the subject and the connections between them. If I have time, I shall say something about the derived commutative algebra of cohomology rings.

Carles Broto: Fusion systems and self equivalences of p-completed classifying spaces of finite groups of Lie type.

This is a talk about the interplay between finite group theory and homotopy theory. We review the concept of fusion system and linking system and how they relate to classifying spaces and self maps. As an application, we show some equivalences between fusion systems of finite groups of Lie type and the relation between self maps of p-completed classifying spaces and fusion preserving automorphisms of Sylow p-subgroups.

Pierre-Emmanuel Caprace: Rank one groups and trees.

We will report on recent progress towards a characterisation of rank one groups over local fields among all locally compact groups, in the spirit of Hilbert's fifth problem. We will explain how trees whose set of ends is a Moufang set appear in that context, and present a classification of those Moufang sets in characteristic zero.

Andrew Chermak: Localities.

Localities are "objective partial groups" which are at the same time "transporter systems" in the sense of homotopical group theory developed by Broto, Levi, Oliver, and their many co-workers. They have provided the basis for a constructive proof of the result that to any saturated fusion system over a finite p-group there corresponds a unique centric linking system. I'll discuss some more recent developments, with a view towards obtaining a suitable notion of "homomorphism" of linking systems, and of "normal subsystem" of a linking system.

Persi Diaconis: Shuffling Cards and Hopf Algebras.

The Hopf-square map on a graded bialgebra often has a natural combinatorial/probabilistic interpretation. On the free associative algebra, it leads to the Gilbert-Shannon-Reeds model of riffle shuffling. Applied to symmetric functions, it leads to a rock-breaking model of Kolmogorov. Using this, the combinatorics of the free algebra and an idea of Drinfeld leads to a description of all the eigen-values and eigen-vectors. These are often natural, useful, and interpretable. This is joint work with Amy Pang and Arun Ram.

Bernd Fischer: Conjugacy-classes of finite groups.

In geometry special properties of collineations have been studied since the 19^th century. In particular involutions were used as subsets of groups to describe geometries. In my work I study conjugacy classes D of involutions generating a finite group G: Let ω be a set of natural numbers; then D is a class of ω-transpositions of G if different elements of D have products of order in ω. I plan to talk on recent results classifying {3,4}-transposition-groups.

Paul Flavell: Signalizer Amalgams.

Let A be an elementary abelian r-group of rank at least 3. We will define the notion of an A-signalizer amalgam. A proof of the classification of the simple A-signalizer amalgams will be described. It is a corollary that every A-signalizer amalgam arises from the action of A on an r'-group. This gives a new proof of McBride's Nonsolvable signalizer Functor Theorem. Consequently, this work will be a contribution to the Gorenstein-Lyons-Solomon proof of the Classification of the Finite Simple Groups.

The proof of the classification of A-signalizer amalgams has many features in common with that of the CFSG, however it is very much simpler.

Terry Gannon: Moonshine old and new.

30 years ago McKay noticed 196883+1 = 196884, and the result was a Moonshine between the Fischer-Griess Monster and the generator j of the modular functions for SL(2,Z). 2 years ago Eguchi-Ooguri-Tachikawa remarked that 90, 462, 1540, 4554, 11592, ... also have group theoretic significance, and another moonshine involving the sporadic groups was born. My talk reviews both of these.

Ellen Henke: Classification Theorems for Fusion Systems.

Saturated fusion systems model the p-local structure of finite groups. They were introduced by Puig in the early 1990's under the name of full Frobenius categories to unify phenomena occurring both in block theory and in finite groups. Later, Broto, Levi and Oliver introduced the now standard terminology and extended Puig's theory for the purposes of homotopy theory. Recent work of Aschbacher and others has enriched the theory of saturated fusion systems by concepts and theorems which have analogues of fundamental importance in finite group theory. In this talk we will focus on classification theorems for fusion systems. In particular, I will outline the current progress of a large project proposed by Michael Aschbacher for the classification of all simple 2-fusion systems leading to a new proof of the classification of finite simple groups. Moreover, I will report about my work towards a classification of minimal non-solvable fusion systems.

Martin Kassabov: Quantifing the residual finiteness of groups.

(joint work with I. Biringer, K Bou-Rabee, and F. Matucci)

A group is called residually finite if the intersection of all finite index subgroups is trivial. This can be quantified in two ways
- what is the minimal index of a normal subgroup, which does not contain a given element; and
- what is the index of the intersection of all normal subgroups of small index.
This leads to new asymptotic invariants of finitely generated residually finite groups.

Radha Kessar: On Brauer's height zero conjecture.

Many questions in the modular representation theory of finite groups concern the connection between global representation-theoretic invariants (e.g. character degrees, cartan numbers, decomposition numbers...) and local structure (e.g. defect groups, fusion systems...). I will speak about the recent resolution of the forward direction of the following problem, posed by Richard Brauer, and known as the height zero conjecture:

The defect groups of a p-block of a finite group are abelian if and only if all irreducible characters in the p-block are of height zero.

This is joint work with Gunter Malle.

Olga Kharlampovich: Definable subsets in a free group.

We give a description of definable subsets in a free non-abelian group F that follows from our work on the Tarski problems. As a corollary we show that proper non-abelian subgroups of F are not definable (solution of Malcev's problem) and prove Bestvina and Feighn's result that definable subsets in a free group are either negligible or co-negligible. This is joint work with A. Myasnikov.

Ian Leary: Platonic polygonal complexes.

I shall define what I mean by a platonic polygonal complex, and state a recent classification result for platonic polygonal complexes with certain vertex links. I shall also explain the connection with incidence geometries and with some earlier work. The original parts of the talk concern joint work with Tadeusz Januszkiewicz, Raciel Valle and Roger Vogeler.

Gunter Malle: From the Weil conjectures to Beauville surfaces, via finite simple groups.

Group theoretic questions can often be reduced to the study of finite simple groups. We show how Lusztig's character theory, which ultimately rests on the Weil conjectures, can be used to obtain strong results on the structure constants of simple groups of Lie type. These in turn lead to the solution of a conjecture of Peter Neumann on fixed spaces in representations, and of a conjecture of Bauer, Catanese and Grunewald on the existence of Beauville surfaces. This is joint work with Robert Guralnick.

Bernhard Mühlherr: Multiple trees.

Around 1989 Ronan and Tits introduced twin buildings which are motivated by the theory of Kac-Moody groups. Twin buildings are pairs of buildings endowed with a codistance function. Twin trees provide a most interesting special case of this theory. In the 1990's Serre pointed out that there is a natural generalisation of twin trees to multiple trees based on the theory of line bundles over rational function fields. The examples arising in this context are at the origin of Ronan's work on the Moufang property of multiple trees.

In joint work with M. Grüninger we proved that multiple Moufang trees involving at least three factors are of algebraic origin (i.e. related to S-arithmetic groups where |S| is the number of factors). By work of Rémy and Ronan one knows that one cannot expect such a result if there are only two factors. In my talk I will present this result and discuss some related results for twin trees and in the higher rank case.

Andrei Rapinchuk: Weakly commensurable groups, with applications to differential geometry.

The notion of weak commensurability was introduced in the ongoing joint work with Gopal Prasad on length-commensurable and isospectral locally symmetric spaces. We have been able to determine when two arithmetic subgroups are weakly commensurable. This leads to various geometric results, some of which are related to the famous question "Can one hear the shape of a drum?"

Aner Shalev: Words and Waring type problems.

Non-commutative analogues of Waring problem in number theory were studied extensively in recent years, where the goal is to express group elements as short products of special elements; these may be powers, commutators, values of a general word w, or elements of a given conjugacy class, or of certain subgroups or subsets. Such problems arise naturally in profinite groups, finite groups, and finite simple groups in particular.

I will describe background, recent results (with various coauthors), and relations to representations, geometry, and growth. I will conclude with some applications and conjectures. The talk will be accessible for a wide audience.

Bernd Stellmacher: From Inside the Cloud: Fischer's Students in Bielefeld in the 70s
(or How I Learned to Inhale Without Smoking).

The talk is mainly about a group of young students from Frankfurt. How they got influenced by Fischer's ideas and came to work with him in Bielefeld. and what some of them made out of it later.

The talk is meant for a general audience of mathematicians, and maybe also for people who got used to the mysteries of life, like pieces of incomprehensible or unexplained terminology.

Jakob Stix: On the section conjecture of anabelian geometry.

Anabelian geometry describes arithmetic and geometry of algebraic curves over number fields in terms of their etale fundamental groups. The section conjecture of Grothendieck in particular, suggests to describe rational points in terms of conjugacy classes of splittings of the fundamental extension. After introducing this anabelian circle of ideas we will show how an abelian approximation to the section conjecture, replacing rational points by 0-cycles of degree 1, relates to an old question of Cassels and Bashmakov. The question is about divisibility properties of elements of the Tate-Shafarevich group inside the Weil-Chatelet group of an abelian variety. The reported results are joint work with Mirela Ciperiani.

Gernot Stroth: ²E₆(2) and F₂ a match made in heaven.

In this talk we will report on some relations between ²E₆(2) and F₂ with supporting actor F₄(2). This starts with the centralizer of a {3,4}-transposition in F₂ and the identification by this centralizer. Then it continues with the geometry given by these transposition, which generalizes to c-extended buildings of F₄-type and their classification. Finally it ends with the study of groups with a large subgroup initiated by U. Meierfrankenfeld, B. Stellmacher and G. Stroth, in which contex ²E₆(2) and F₂ show up as honorary groups of Lie type over F₃.
pdf

John Thompson: SL(2) and its quotient groups.

Franz Timmesfeld: Parapolar spaces and the large sporadic groups.

Parapolar spaces are point line geometries with main axiom: if one takes two points of distance two, than there exists either a unique common neighbour or the set of common neighbours is a polar space. In the talk parapolar spaces with 3 points on each line will be considered, for which the exceptional Lie-type groups over GF(2) and certain large sporadic groups (for example monster and baby monster) with point set the set of 2-central involutions provide examples.

Anatoly Vershik: "Random subgroups" and representation theory.
(The measures on the lattice of the subgroups of a given group which are invariant under conjugations.)

Let G be a countable group and L(G) denotes the lattice of all subgroups of G. The group G acts on L(G) by conjugations. Are there nontrivial diffusive probability measures (i.e., without atoms) on L(G) which are invariant under conjugations? This question appears in the context of the infinite dimensional representations of the group. A complete answer was obtained for the infinite symmetric group and it is deeply connected to the Thoma theorem about the list of characters of that group.

Karen Vogtmann: Hairy graphs, automorphisms of free groups and modular forms.

The group of outer automorphisms of a free group acts on a space of finite graphs known as Outer space, and a classical theorem of Hurwicz implies that the homology of the quotient by this action is an invariant of the group. A more recent theorem of Kontsevich relates the homology of this quotient to the Lie algebra cohomology of a certain infinite-dimensional symplectic Lie algebra. Using this connection, S. Morita discovered a series of new homology classes for Out(F_n). In joint work with J. Conant and M. Kassabov, we reinterpret Morita's classes in terms of hairy graphs, and show how this graphical picture then leads to the construction of large numbers of new classes, including some based on classical modular forms for SL(2,Z).

Richard Weiss: Three families of exceptional groups.

Like the three Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄, the Moufang quadrangles "of type E₆, E₇ and E₈" arise inductively from one special case in the solution to a larger classification problem. We will describe how this happens and try to indicate why this induction, like Fischer's, stops after just three steps. We will also discuss various properties of these remarkable geometries and their automorphism groups. For example, if G is the group of linear automorphisms of one of these quadrangles, then F*(G) is simple (but not finite); these simple groups are the exceptional groups of our title.

Robert A Wilson: Simple groups of Lie type without Lie theory.

While Lie theory has been very successful at unifying the theory of finite simple groups, its size and complexity present a barrier to beginning students. Moreover, it begins from the adjoint representation, which in general is not the smallest representation, and is not a representation of the generic cover.

Alternative approaches can give a more direct construction, overcoming all of these obstacles, though at the cost of uniformity. This is well-known in the case of classical groups: we present some new approaches to exceptional groups.