Trigonometric chaos and $\mathrm{X}_p$ inequalities I -- Balanced Fourier truncations over discrete groups

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.