  We investigate Fourier multipliers on the compact dual of arbitrary discrete
groups. Our main result is a H\"ormander-Mihlin multiplier theorem for
finite-dimensional cocycles with optimal smoothness condition. We also find
Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$
estimates for noncommutative Riesz transforms and characterize $L_\infty \to
\mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of
our approach are to exploit group cocycles and cross products in Fourier
multiplier theory in conjunction with BMO spaces associated to semigroups of
operators and a noncommutative generalization of Calder\'on-Zygmund theory.
