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Description

I will survey a few papers concerning the following question: What can one learn about $G$ from its representation theory into $S$. Where $S=S_\infty$ is the Polish group of all permutations of a countable set. $\text{Hom}(G,S)$ is a Polish space in its own right. We focus on two aspects. How a (Baire) generic representation of $G$ looks like. And the existence of representations with special transitivity properties such as faithful primitive actions, or faithful highly transitive actions of the group.