Group Action Forum

Groups are fundamental objects in modern mathematics, and groups are useful because they act on appropriate spaces. Group actions allow us to understand both spaces and groups. There has been a lot of recent work on applications of group actions in geometry, topology and related subjects. For example, various rigidity properties of group actions have been pursued by people and include rigidity of Tits buildings by Tits, strong rigidity of locally symmetric spaces by Mostow, superrigidity on lattices of semisimple Lie groups by Margulis, rank rigidity of nonositively curved manifolds by Ballmann and Burns-Spatzier, and the most recent rigidity on polar actions by Fang-Grove-Thorbergsson and Lytchak. Group actions of infinite discrete groups such as mapping class groups and outer automorphism groups of free groups have motivated much of study in geometric group theory, which is essentially the subject of studying groups by considering actions on geometric and topological spaces. There are different types of groups such as discrete groups and continuous (or Lie) groups, and their actions have often been studied separately from each other. The time seems to be ripe to organize a workshop with participation by leading experts around the work on various aspects of group actions. Besides some survey talks on geometrical and topological applications, we also hope to organize some more introductory talks suitable to beginners and nonexperts. Such a workshop will make people more aware of the importance of studying these different groups together (as witnessed in the classical study of the pairs of groups (Z, R)) and lead to more interaction between these different camps of people and possible collaboration. More importantly, it will attract the younger generation to this elegant and important subject. Though group actions are important, there have not been detailed and systematic descriptions of groups. We plan to edit a book (or books) on group actions with contribution by the participants. They will consist of introductions and survey papers. Such books should be valuable to both beginners and experts.

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Fu-Quan FangCapital Normal University, China
Lizhen JiUniversity of Michigan
Athanase PapadopoulosUniversity of Strasbourg
Shing-Tung YauHarvard University