  For locally compact groups amenability and Kazhdan's property (T) are
mutually exclusive in the sense that a group having both properties is compact.
This is no longer true for more general Polish groups. However, a weaker result
still holds for SIN groups (topological groups admitting a basis of
conjugation-invariant neighbourhoods of identity): if such a group admits
sufficiently many unitary representations, then it is precompact as soon as it
is amenable and has the strong property (T) (i.e. admits a finite Kazhdan set).
If an amenable topological group with property (T) admits a faithful uniformly
continuous representation, then it is maximally almost periodic. In particular,
an extremely amenable SIN group never has strong property (T), and an extremely
amenable subgroup of unitary operators in the uniform topology is never a
Kazhdan group. This leads to first examples distinguishing between property (T)
and property (FH) in the class of Polish groups. Disproving a 2003 conjecture
by Bekka, we construct a complete, separable, minimally almost periodic
topological group with property (T), having no finite Kazhdan set. Finally, as
a curiosity, we observe that the class of topological groups with property (T)
is closed under arbitrary infinite products with the usual product topology.
