A fundamental dichotomy in the theory of infinite groups is the one between amenable groups and groups with Kazhdan's Property (T). In this talk I shall overview versions of these two opposite properties, connections to actions on non-positively curved spaces and on Banach spaces, to other geometric features of the groups, and to expander graphs. I shall also mention what is known in the setting of random groups and that of important classes of infinite groups (e.g. lattices, mapping class groups, Out(F_n) etc)

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