Groups are core to algebra, and their study now covers a wide range of techniques. Modern advances in group theory utilise categories to study properties for finite and infinite groups alike. Results obtained using categories such as fusion systems have allowed significant progress in the local-global study of finite groups, while Mackey functors and Bredon cohomology have been a major feature of the functorial study of groups, leading to major advances also in neighbouring areas such as algebraic topology, representation theory and K-theory in particular.
These functorial techniques, mainly developed for finite groups to date, have emerged in the study of infinite groups, and more recently in the study of profinite groups. This Research Group aims to bring together researchers representing the various subjects touched by functor categories for groups in order to incentivise future advances and stimulate new collaborations in the UK and Ireland.
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The next meeting will be
- Cohomology and Mackey Functors for Profinite Groups
December 2019, Royal Holloway, University of London; to be held at Senate House, Central London (local organiser Brita Nucinkis)
This will be the final meeting in 2019 of the Research Group Functor Categories for Groups (FCG). Speakers at this meeting are N. Mazza (Lancaster), G. Corob Cook (Bilbao) and T. Weigel (Milano).
- Words in finite and profinite groups
September 2019, Lincoln (local organiser Anitha Thillaisundaram)
This meeting shall focus on a natural object in group theory: words. Several classical topics that are closely linked to the study of words are varieties of groups and group laws. Words also give rise to the concept of a verbal subgroup and is important in the study of group laws. The study of words has made significant contributions to our understanding of finite groups as well as profinite groups, as seen for example, by the influential theorem of Segal and Nikolov, that a finite index subgroup of a finitely generated profinite group is open.
- June 2019
Cambridge (local organiser Rachel Camina)
- Cohomology of functor categories for infinite discrete groups
May 2017, Galway, Ireland (local organiser Dieter Degrijse)
The meeting shall focus on applications of functor cohomology and cohomology in categories to the study of infinite discrete groups focusing in particular on recent applications to homological stability and connections with equivariant stable homotopy theory, finiteness properties of groups, Mackey functors and Bredon cohomology. This meeting will be integrated into the 2017 Groups in Galway conference.
- (Pro-)fusion systems
September 2017, Lancaster (local organiser Nadia Mazza)
Introduced in the '70s, fusion systems are categories which model how non-conjugate subgroups in a Sylow p-subgroup of a given finite group can fuse, i.e. become conjugate, in the whole group. The study of fusion systems has led to significant advances and improvements of proofs in group theory, and also provided useful links with algebraic topology. The focus of the meeting will be on the use of fusion systems in the local to the global theory of finite groups and the theory of profinite groups.
- The category of totally disconnected locally compact groups
April 2018, Lincoln (local organiser Anitha Thillaisundaram)
This meeting, which is more of a masterclass in nature, shall focus on the category of totally disconnected locally compact groups and how they intersect with other areas, such as permutation groups, operator algebras and model theory. The totally disconnected compact groups are of course the profinite groups, whose influence is far-reaching. The locally-compact case has received much attention since the ground-breaking result of Willis in 1994.
- Cohomology of functor categories for topological groups
September 2018, London (local organiser Brita Nucinkis)
We shall focus on the study of Hausdorff dimension for profinite groups, initiated by Abercrombie in the 90s.
- Graphs and groups
December 2018, Lancaster (local organiser Nadia Mazza)
We shall focus on the interplay of graphs and groups, and how each of these structures can be used in the study of the other. Groups act on varied combinatorial structures and graphs in particular. On the other hand, given an abstract group, there are several ways to construct a graph associated to the group, such as π-products involution graphs, or commuting graphs. Recently, the approach of studying groups via associated graphs has been extended to hypergraphs, which leads to a useful generalisation of the theory.
- Stable categories
April 2019, Lancaster (local organiser Nadia Mazza)
This meeting shall focus on stable (module) categories in group representation theory. There are different notions of "stable categories", but all have a common point: they are quotient categories, which are triangulated and have an extra multiplicative structure with a multiplicative identity. Stable categories are also intrinsically related to derived categories. The selected speakers are experts in varied aspects of stable and derived categories, and some of their recent results exhibit useful applications of stable categories.
The Research Group receives financial support from the London Mathematical Society and has, therefore, limited funds to reimburse travel expenses of UK-based students and young mathematicians. Please contact the organisers if you wish to apply for such reimbursements.
For UK-based mathematicians with caring duties, the LMS has a Caring Supplementary Grant scheme which allows participants of meetings like ours to apply for help covering caring costs.
Corresponding organisers of the Joint Research Group are: