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

Metric groups, unitary representations and continuous logic