  We describe how properties of metric groups and of unitary representations of
metric groups can be presented in continuous logic. In particular we find
$L_{\omega_1 \omega}$-axiomatization of amenability. We also show that in the
case of locally compact groups some uniform version of the negation of
Kazhdan's property {\bf (T)} can be viewed as a union of first-order
axiomatizable classes. We will see when these properties are preserved under
taking elementary substructures.
