### Addendum to "A survey of semigroups of continuous selfmaps" (with intro. by K. D. Magill).

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Let $G{=}_{\infty}\left(T\right)$ be the wreath product of a compact group T with the infinite symmetric group ${}_{\infty}$. We study the characters of factor representations of finite type of G, and give a formula which expresses all the characters explicitly.

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation...

We show the irreducibility of some unitary representations of the group of symplectomorphisms and the group of contactomorphisms.

We show that every abelian Polish group is the topological factor group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.