Galaxy cluster outskirts are described by complex velocity fields induced by diffuse material collapsing towards filaments, gas and galaxies falling into clusters, and gas shock processes triggered by substructures. A simple scenario that describes the large-scale tidal fields of the cosmic web is not able to fully account for this variety, nor for the differences between gas and collisionless dark matter. We have studied the filamentary structure in zoom-in resimulations centred on 324 clusters from The ThreeHundred project, focusing on differences between dark and baryonic matter. This paper describes the properties of filaments around clusters out to five $R_{200}$, based on the diffuse filament medium where haloes had been removed. For this, we stack the remaining particles of all simulated volumes to calculate the average profiles of dark matter and gas filaments. We find that filaments increase their thickness closer to nodes and detect signatures of gas turbulence at a distance of $sim 2 h^{-1}rm{Mpc}$ from the cluster. These are absent in dark matter. Both gas and dark matter collapse towards filament spines at a rate of $sim 200 h^{-1} rm{km ~ s^{-1}} $. We see that gas preferentially enters the cluster as part of filaments, and leaves the cluster centre outside filaments. We further see evidence for an accretion shock just outside the cluster. For dark matter, this preference is less obvious. We argue that this difference is related to the turbulent environment. This indicates that filaments act as highways to fuel the inner regions of clusters with gas and galaxies.
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