Y. Cornulier

16 - 18 April

Title: Near actions.

Abstract:

A near action (of a group on a set) is defined similarly as an action, but  "modulo indeterminacy on finite subsets". The realizability issues consisting in investigating obstructions for near actions to be induced by actions.

While the notion of near actions notion is new, it appears in several known settings. I will introduce this notion in detail in a first time, and then will make several independent lectures (depending on the audience's wish) on topics in which this notion naturally occurs:

- generalities on near actions of finitely generated groups
- classification of near actions of finitely generated abelian groups
- the near action of the group of interval exchanges with flips on the circle
- near actions of SL(2,Z) and amalgams of finite groups
- locally finite groups, and maximal abelian subgroups of the quotient of the symmetric group by the subgroup of finitary permutations
- the Kapoudjian 2-cocycle
- rigidity for near actions of 1-ended groups
- near automorphism groups (such as Thompson's groups and Neretin's groups).

I'm also planning a more informal lecture on the history of infinite symmetric groups, which, up to my knowledge, starts with a 1915 note by Vitali showing that infinite symmetric groups have no "signature", that is, no nontrivial homomorphism onto the cyclic group of 2 elements.

This should be of interest to people with an interest in infinite permutation groups, in Polish groups, and in abstract groups dynamics.