  We show that all groups of a distinguished class of \guillemotleft
large\guillemotright\ topological groups, that of Roelcke precompact Polish
groups, have Kazhdan's Property (T). This answers a question of Tsankov and
generalizes previous results by Bekka (for the infinite-dimensional unitary
group) and by Evans and Tsankov (for oligomorphic groups). Further examples
include the group $\operatorname{Aut}(\mu)$ of measure-preserving
transformations of the unit interval and the group $\operatorname{Aut}^*(\mu)$
of non-singular transformations of the unit interval.
  More precisely, we prove that the smallest cocompact normal subgroup
$G^\circ$ of any given non-compact Roelcke precompact Polish group $G$ has a
free subgroup $F\leq G^\circ$ of rank two with the following property: every
unitary representation of $G^\circ$ without invariant unit vectors restricts to
a multiple of the left-regular representation of $F$. The proof is
model-theoretic and does not rely on results of classification of unitary
representations. Its main ingredient is the construction, for any
$\aleph_0$-categorical metric structure, of an action of a free group on a
system of elementary substructures with suitable independence conditions.
