  In the line of previous work by Naor, we establish new forms of metric
$\mathrm{X}_p$ inequalities in group algebras under very general assumptions.
Our results' applicability goes beyond the previously known setting in two
directions. In first place, we find continuous forms of the $\mathrm{X}_p$
inequality in the $n$-dimensional torus. Second, we consider transferred forms
of the sharp scalar valued metric $\mathrm{X}_p$ inequality in the von Neumann
algebra $\mathcal{L}(\mathrm{G})$ of a discrete group $\mathrm{G}$. As a
byproduct of our results, some metric consequences and their relation with
bi-Lipschitz nonembeddability of Banach spaces are explored in the context of
noncommutative $L_p$ spaces.
