Geometric group theory beyond groups

In this talk I will first explain the basic ideology behind geometric group theory: How and to what extend can we understand (finitely-generated) groups as geometric objects?  I will discuss the classical Schwarz-Milnor lemma which provides a translation mechanism between groups and geometry. In particular I will discuss a certain class of isometric actions called geometric actions. I will then explain that the Schwarz-Milnor machinery not only applies to isometric actions, but also to quasi-isometric quasi-actions of groups, and try to convince you that this is actually the more natural context of modern geometric group theory. 

In the final part of my talk, I will discuss some very recent developments which show that one can not only “quasify" the notion of an isometric action but also the notion of a group itself. This allows us to not only interpret groups, but also more general algebraic structures called approximate groups as geometric objects. Time permitting I will comment on various algebraic, geometric and analytic aspects of approximate groups. This final part is based on joint work with Michael Björklund and Matthew Cordes. 

 No prior knowledge of geometric group theory is required and large parts of the talk should be understandable to master and PhD students.