Michal DOUCHA: Combinatorial group theory in the metric setting

ABSTRACT: There are two main ways how to define discrete groups. Either as the set of symmetries
of some geometric object, or in terms of generators and relations they satisfy. The latter
approach is studied by the classical combinatorial group theory.

Our aim is to present some results that use methods of combinatorial group theory generalized
into the setting of metric/topological groups. Researchers working with Polish groups usually 
only consider examples that are automorphisms of some topological or metric structure.
We shall present some Polish metric groups that are constructed in terms of generators and metric 
inequalities these generators must satisfy. Namely, we present some universal Polish groups 
constructed in this way, whose existence had been open before, and the first example of
a non-commutative uniform Banach group, i.e. a Banach space with an additional group structure
which is uniformly continuous in the norm topology. Finally, we explain how this "metric
combinatorial group theory" might be related to metric approximation of groups, a concept that
is famous mainly because of notions such as sofic and hyperlinear groups.