# Groups and Geometries (15w5017)

Arriving in Banff, Alberta Sunday, May 3 and departing Friday May 8, 2015

## Organizers

Inna Capdeboscq (University of Warwick, UK)

(Imperial College London, UK)

Bernhard Muehlherr (University of Giessen, Germany)

## Objectives

We propose a meeting focussing on three main themes. The first is the theory of fusion systems. This has grown into a new field in finite group theory where exciting developments are currently happening. The formalized study of fusion systems was started by Puig and based on the work of Alperin and Brou'e in representation theory on $p$-blocks, defect groups and Brauer pairs. This led Puig to the abstract notion of a fusion system as a category in 1992. The work of Benson, Broto and Levi showed that there are fusion systems which do not belong to groups. The groundbreaking work of Broto, Levi and Oliver in 2003, which set up the foundation of the homotopy theory in the area of fusion systems was a new launch for the field. Their key result is a cohomological criterion on whether there is a topological space associated with a fusion system on a $p$-group, which coincides with the $p$-completed classifying space of a finite group $G$ such that the fusion system is the one of this group. Oliver used this criterion to answer the long standing Martino-Priddy conjecture that two groups have homotopy equivalent classifying spaces if and only if they have isomorphic fusion systems. Recently Chermak showed that any fusion system has such a space. Fusion systems is a fast growing area which provides a link between finite group theory, modular representation theory and homotopy theory. Some of the new results obtained in this area answer natural questions in finite group theory that could have been formulated a long time ago; but the tools were not developed to answer them. M. Aschbacher in fact is trying to extend local group theory to fusion systems. In the 2012 meeting that took place in BIRS, M. Aschbacher proposed a programme for classifying simple $2$-fusion systems of component type. If it is successfully carried out, one of the outcomes might come into resonance with the MSS (Meierfrenkenfield, Stellmacher, Stroth)-programme for the revision of parts of the classification of finite simple groups. The combination of both projects (that of M. Aschbacher and the MSS-programme) would provide a possible classification of all simple $2$-fusion systems. Chermak introduced in the work mentioned above a completely novel structure, the so-called objective partial groups. This might be a key to developing an equivalent of the amalgam method in the context of fusion systems. The amalgam method proved to be one of the most fruitful tools developed during the revision of the classification of the finite simple groups and is at the core of the MSS-progamme. This might also be what is needed to connect the MSS-programme to the Aschbacher-programme. As a consequence, this might result in a new proof of the classification of the finite simple groups. This connection between the two projects was rather unexpected. The response to M. Aschbacher's new programme has been very enthusiastic, and at the same time the MSS-programme is progressing at full speed. This leads us to expect new developments in finite group theory in the next years, in particular in fusion systems. We believe that there will be substantial progress in both programmes and this will be a perfect moment to discuss how close we are to the classification of the simple $2$-fusion systems. The second theme concerns the connections between finite simple groups and semi-simple algebraic groups on the one side and combinatorial geometry, in particular buildings, on the other side. This interplay has a fruitful history and remains a source of inspiration for both areas. In fact buildings were created by Tits about 50 years ago in order to study groups of Lie type from a combinatorial point of view. His programme has been most successful and it still is. This is underlined by recent applications of building theory to the MSS-programme mentioned before. Here, the celebrated Local-to-Global-Theorem for spherical buildings is used to overcome a major obstacle in the endgame of the MSS-project. In the other direction, a variation of the classical Baer-Suzuki-Theorem due to Aschbacher and Timmesfeld has been used recently to give a short and conceptual proof of the centre conjecture for spherical buildings. This conjecture has been open for about 50 years and regained interest through the work of Serre on complete reducibility. The earlier case-by-case proof had been accomplished by Ramos-Cuevas in 2009 by treating buildings of type $E_8$ in a long and difficult paper. Perhaps more important than these direct applications from one field to the other is that both group theory and geometry provide each other with interesting and challenging problems. One outstanding example for this is the theory of rank 1 groups, of which the finite ones play a central role in the context of finite simple groups. The infinite rank 1 groups will constitute a major theme at the conference, since there are currently exciting developments going on. Whereas earlier results on rank 1 groups were mostly obtained by applying techniques from abstract group theory and Jordan algebras, the current contributions have a more geometric flavour. There are new results and ongoing research on the interplay between groups acting on trees and rank 1 groups acting on the set of ends. The most spectacular result, due to Caprace and De Medts, is on classifying Moufang sets which occur as the set of ends of a locally finite tree in characteristic 0. Exceptional groups and their geometries will be another focus of the conference. The Freudenthal-Tits Magic Square provides the right setting for approaching the exceptional spherical buildings in alternative ways, such as via algebraic varieties in the sense of Mazzocaa-Melone, or via incidence geometries a la Klingenberg and Hjelmslev. Both ways are promising to find an explicit construction of the $E_8$ geometry in 247 (projective) dimensions, the ultimate goal of a programme which was presented by Van Maldeghem during the 2012 meeting 'Groups and Geometries' in BIRS. In the area of exceptional groups there is also ongoing work of Weiss on Bruhat-Tits buildings of exceptional types which reveals interesting connections with the theory of pseudo-reductive groups whose theory has been developed recently by Conrad, Gabber and Prasad. The third theme concerns applications of finite group theory, in particular the classification of simple groups, to problems both within and outside group theory. This area is very diverse, and we discuss just one example, coming from algebraic geometry. According to a definition of M. Reid, if $V$ is a vector space of dimension $d$ over the complex numbers, the age of a matrix in the unitary group $GU(V)$ having eigenvalues ${rm exp(2pi ir_i)$ (where $0 le r_i< 1$, $i = 1,ldots ,d$) is defined to be the sum of the exponents $r_i$. A result of Ho and Reid states that if $G$ is a finite subgroup of the unitary group, and if the variety $V/G$ admits a crepant resolution, then $G$ contains an element of age less than 1. Thus the question arises whether finite group theory can be used to say something about such finite subgroups. Partial results on this and many related questions have been obtained by Guralnick, Larsen, Tiep and others, and used by Kollar, Reid and others to prove results in the theory of resolutions of algebraic varieties in algebraic geometry. There seems to be great scope for further such applications of finite group theory in this direction.