We investigate the automorphism groups of $\aleph\_0$-categorical structuresand prove that they are exactly the Roelcke precompact Polish groups. We showthat the theory of a structure is stable if and only if every Roelcke uniformlycontinuous function on the automorphism group is weakly almost periodic.Analysing the semigroup structure on the weakly almost periodiccompactification, we show that continuous surjective homomorphisms fromautomorphism groups of stable $\aleph\_0$-categorical structures to Hausdorfftopological groups are open. We also produce some new WAP-trivial groups andcalculate the WAP compactification in a number of examples.
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