We report x-ray diffraction, magnetic susceptibility, heat capacity, $^{1}$H nuclear magnetic resonance (NMR), and muon spin relaxation ($mu$SR) measurements, as well as density-functional band-structure calculations for the frustrated $S=3/2$ triangular lattice Heisenberg antiferromagnet (TLHAF) $alpha$-HCrO$_{2}$ (trigonal, space group: $Rbar{3}m$). This compound undergoes a clear magnetic transition at $T_{rm N} simeq 22.5$~K, as seen from the drop in the muon paramagnetic fraction and concurrent anomalies in the magnetic susceptibility and specific heat. Local probes (NMR and $mu$SR) reveal a broad regime with slow fluctuations down to $0.7,T_{rm N}$, this temperature corresponding to the maximum in the $mu$SR relaxation rate and in the NMR wipe-out. From the comparison with NaCrO$_{2}$ and $alpha$-KCrO$_{2}$, the fluctuating regime and slow dynamics below $T_{rm N}$ appear to be hallmarks of the TLHAF with $ABC$ stacking that leads to a frustration of interlayer couplings between the triangular planes. This interlayer frustration is a powerful lever to generate spin states with persistent dynamics and may bear implications to spin-liquid candidates with the triangular geometry.
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