University of Toronto's Symplectic Geometry Seminar

Abstract: We say that a topological group $G$ is $n$-bounded if it contains a ``small" subset $K$ such that every element $g$ in $G$ can be represented as a product of at most $n$ elements, each of which is conjugate to some element in $K$. Here is our original motivation for this definition: Even though the entire theory of dynamical systems studies properties of maps that are invariant under conjugations, these are very specific properties (usually related to iterations of maps in question). For instance, these properties usually get completely destroyed by compositions of maps. After looking at certain stability questions in sequential dynamics, we formulated this definition as a ``test tool'' for looking for more ``geometric'' invariants (of conjugacy classes) which do not get completely destroyed by compositions. There is a well-known remarkable invariant of this type: Hofer's norm. If a transformation group is unbounded, this means that there is necessarily a certain invariant responsible for that. Now it also seems that the property of being (un)bounded also can be useful by itself (for instance, the image of a representation of a bounded group lies in a bounded subgroup). Whereas for some groups it is very easy to see if they are bounded or not, in many cases this appears to be a rather difficult question. We will discuss some cases where we were able to answer this question, and many examples where we are so far unable to answer it. The talk is based on a joint project with S. Ivanov and L. Polterovich.