Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

We investigate the automorphism groups of $\mathrm{\aleph }\mathrm{\_}0$-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable $\mathrm{\aleph }\mathrm{\_}0$-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and calculate the WAP compactification in a number of examples.