## Cornelia Druțu (Oxford U):
Actions on Non-positively Curved Spaces and on Banach Spaces and Geometry of Infinite (Random) Groups

One way of understanding infinite discrete groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc. Classical properties such as Kazhdan's property (T), the Haagerup property, and their recent generalizations, are formulated in terms of such actions and turn out to be relevant in a wide range of areas, from the construction of expanders to the Baum-Connes conjecture.
A source of interesting examples comes from the theory of random groups, a close relative of the theory of random graphs. Random groups are Gromov hyperbolic and have property (T). Moreover, for generic random groups there is a connection between the conformal dimension of their boundary and fixed point properties for affine isometric actions on uniformly curved Banach spaces. On the other hand, there exist examples of exotic hyperbolic groups for which no such connection can be established.
In my talk I shall overview key results on the above topics and some recent developments.