Large scale geometry of topological groups

Abstract: Geometric group theory or more precisely large scale geometry of discrete groups is based on the fundamental observation that the word metrics on a discrete group given by distinct finite generating sets are bi-Lipschitz equivalent, i.e., differ at most by a multiplicative constant. This observation makes it possible to treat finitely generated groups as geometric objects as long as the methods employed are insensitive to the multiplicative error and has led to a very rich interplay between numerous mathematical disciplines such as algebra, topology, functional and harmonic analysis, ergodic theory and logic. Moreover, this study carries quite easily over to compactly generated locally compact groups, but so far more general topological transformation groups, e.g., homeomorphism and diffeomorphism groups, have resisted treatment from this perspective due to the presumed absence of canonical generating sets. We shall present some newly developed tools for overcoming this. The talk will be aimed at a general audience.