Arbitrating the $S_8$ discrepancy with growth rate measurements from
  Redshift-Space Distortions
  Within the $\Lambda$CDM model, measurements from recent Cosmic Microwave
Background (CMB) and weak lensing (WL) surveys have uncovered a $\sim 3\sigma$
disagreement in the inferred value of the parameter $S_8 \equiv
\sigma_8\sqrt{\Omega_m/0.3}$, quantifying the amplitude of late-time matter
fluctuations. Before questioning whether the $S_8$ discrepancy calls for new
physics, it is important to assess the view of measurements other than CMB and
WL ones on the discrepancy. Here, we examine the role of measurements of the
growth rate $f(z)$ in arbitrating the $S_8$ discrepancy, considering
measurements of $f\sigma_8(z)$ from Redshift-Space Distortions (RSD). Our
baseline analysis combines RSD measurements with geometrical measurements from
Baryon Acoustic Oscillations (BAO) and Type Ia Supernovae (SNeIa), given the
key role of the latter in constraining $\Omega_m$. From this combination and
within the $\Lambda$CDM model we find $S_8 = 0.762^{+0.030}_{-0.025}$, and
quantify the agreement between RSD+BAO+SNeIa and \textit{Planck} to be at the
$2.2\sigma$ level: the mild disagreement is therefore compatible with a
statistical fluctuation. We discuss combinations of RSD measurements with other
datasets, including the $E_G$ statistic. This combination increases the
discrepancy with \textit{Planck}, but we deem it significantly less robust. Our
earlier results are stable against an extension where we allow the dark energy
equation of state $w$ to vary. We conclude that, from the point of view of
combined growth rate and geometrical measurements, there are hints, but no
strong evidence yet, for the \textit{Planck} $\Lambda$CDM cosmology
over-predicting the amplitude of matter fluctuations at redshifts $z \lesssim
1$. From this perspective, it might therefore still be premature to claim the
need for new physics from the $S_8$ discrepancy.
