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Context. We have studied the dense gas morphology and kinematics towards the infrared dark cloud (IRDC) G034.77-00.55, shock-interacting with the SNR W44, to identify evidence of early-stage star formation induced by the shock. Aims. We used high angular resolution N₂H⁺(1−0) images across G034.77-00.55, obtained with the Atacama Large Millimeter/sub-Millimeter Array. N₂H⁺is a well-known tracer of dense and cold material, optimal for identifying gas that has the highest potential to harbour star formation. Methods. The N₂H⁺emission is distributed in two elongated structures, one towards the dense ridge at the edge of the source and one towards the inner cloud. Both elongations are spatially associated with well-defined mass-surface density features. The velocities of the gas in the two structures (i.e. 38–41 km s⁻¹and 41–43 km s⁻¹) are consistent with the lowest velocities of the J- and C-type parts, respectively, of the SNR-driven shock. A third velocity component is present at 43–45.5 km s^–1. The dense gas shows a fragmented morphology with core-like fragments at scales consistent with the Jeans lengths, masses of ~1–20 M_⊙, densities of (n(H₂)≥10⁵cm^–3) sufficient to host star formation in free-fall timescales (a few 10⁴yr), and with virial parameters that suggest a possible collapse. Results. The W44 driven shock may have swept up the encountered material, which is now seen as a dense ridge, almost detached from the main cloud, and an elongation within the inner cloud, well constrained in both N₂H⁺emission and mass surface density. This shock compressed material may have then fragmented into cores that are either in a starless or pre-stellar stage. Additional observations are needed to confirm this scenario and the nature of the cores.