  If $G$ is a Polish group and $\Gamma$ is a countable group, denote by
$\Hom(\Gamma, G)$ the space of all homomorphisms $\Gamma \to G$. We study
properties of the group $\cl{\pi(\Gamma)}$ for the generic $\pi \in
\Hom(\Gamma, G)$, when $\Gamma$ is abelian and $G$ is one of the following
three groups: the unitary group of an infinite-dimensional Hilbert space, the
automorphism group of a standard probability space, and the isometry group of
the Urysohn metric space. Under mild assumptions on $\Gamma$, we prove that in
the first case, there is (up to isomorphism of topological groups) a unique
generic $\cl{\pi(\Gamma)}$; in the other two, we show that the generic
$\cl{\pi(\Gamma)}$ is extremely amenable. We also show that if $\Gamma$ is
torsion-free, the centralizer of the generic $\pi$ is as small as possible,
extending a result of King from ergodic theory.
