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Escola de Altos Estudos: My Favorite Groups

Professor: Étienne Ghys

The world of groups is vast and meant for wandering! I will give eight talks describing eight groups, or class of groups, that I find fascinating. These talks will be independent and I’ll have no intention to be exhaustive (this would be silly since there are uncountably many groups, even finitely generated!).

In each talk, I will introduce the hero, state one or two results, and formulate one or two conjectures.

Here is a list of possible candidates (I might change it slightly, depending on the wishes of the audience).

Finite groups and their actions on manifolds. I will discuss a theorem by Jordan on projective actions and recent progresses on a conjecture related to actions by diffeomorphisms. The group of diffeomorphisms of the circle is a huge source of open questions. I will discuss its (Gelfand-Fuchs) cohomology and describe standard conjectures related to cobordism of foliations.

In a similar way, the algebraic structure of the group of area preserving diffeomorphisms (or homeomorphisms) of surfaces is still a great mystery, even though amazing progresses have been obtained recently.

The Neretin group is some kind of analog of the group of diffeomorphisms in the p-adic category. I will describe some of its algebraic property and suggest some questions on the associated dynamical systems.

Lie groups are of course fundamental objects. I would like to dedicate a lecture to the ax+b group and its actions on 3-manifolds. This will be an opportunity to describe the recent work of Asaoka.

Thompson’s group is intriguing and can be thought as a discrete subgroup of the group of diffeomorphisms of the circle. One still does not know whether or not it is amenable. Progresses on this question have been chaotic recenty.

Lattices in semi-simple groups have been studied for 150 years. I will restrict myself to the description of their actions on the circle and the corresponding conjecture on their orderability.Gromov created the theory of random groups. I will give a very general overview of this theory which is a very convenient tool to construct counter-examples.
Professor: Étienne Ghys

The world of groups is vast and meant for wandering! I will give eight talks describing eight groups, or class of groups, that I find fascinating. These talks will be independent and I’ll have no intention to be exhaustive...
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