  We investigate $L_p$-estimates for balanced averages of Fourier truncations
in group algebras, in terms of differential operators acting on them. Our
results extend a fundamental inequality of Naor for the hypercube (with
profound consequences in metric geometry) to discrete groups. Different
inequalities are established in terms of directional derivatives which are
constructed via affine representations determined by the Fourier truncations.
Our proofs rely on the Banach $\mathrm{X}_p$ nature of noncommutative
$L_p$-spaces and dimension-free estimates for noncommutative Riesz transforms.
In the particular case of free groups we use an alternative approach based on
free Hilbert transforms.
