Every fixed-point-free isometric action of a semisimple Lie group G is proper.

I will explain this in my talk. Later I will elaborate on further generalizations of this fact.

For example, the image of G under any continuous homomorphism into a topological group is closed. Another application is given by describing explicitly the WAP compactification of G (a notion that will be defined in the talk). The latter will be related to the decay property of matrix coefficients of reflexive representations.

The talk will be based on a joint work with Tsachik Gelander