  We report flow measurements in rotating Rayleigh--B\'enard convection in the
rotationally-constrained geostrophic regime. We apply stereoscopic particle
image velocimetry to measure the three components of velocity in a horizontal
cross-section of a water-filled cylindrical convection vessel. At a constant,
small Ekman number $Ek=5\times 10^{-8}$ we vary the Rayleigh number $Ra$
between $10^{11}$ and $4\times 10^{12}$ to cover various subregimes observed in
geostrophic convection. We also include one nonrotating experiment. The scaling
of the velocity fluctuations (expressed as the Reynolds number $Re$) is
compared to theoretical relations expressing balances of
viscous--Archimedean--Coriolis (VAC) and Coriolis--inertial--Archimedean (CIA)
forces. Based on our results we cannot decide which balance is most applicable
here; both scaling relations match equally well. A comparison of the current
data with several other literature datasets indicates a convergence towards
diffusion-free scaling of velocity as $Ek$ decreases. However, the use of
confined domains leads at lower $Ra$ to prominent convection in the wall mode
near the sidewall. Kinetic energy spectra point at an overall flow organisation
into a quadrupolar vortex filling the cross-section. This quadrupolar vortex is
a quasi-two-dimensional feature as it only manifests in energy spectra based on
the horizontal velocity components. At larger $Ra$ the spectra reveal the
development of a scaling range with exponent close to $-5/3$, the classical
exponent for inertial-range scaling in three-dimensional turbulence. The
steeper $Re(Ra)$ scaling at low $Ek$ and development of a scaling range in the
energy spectra are distinct indicators that a fully developed, diffusion-free
turbulent flow state is approached, sketching clear perspectives for further
investigation.
