# University of Toronto's Symplectic Geometry Seminar

Nov. 11 2002, 2:10pm

SS5017A

## Dmitri Burago

###
Penn. State University

##
``Search for geometric properties of conjugacy classes'' and
unbounded quasi-semi-norms and groups

** 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.