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In the high-Tc cuprates, the unusual dependence of Tc on external pressureresults from the combination of the nonmonotonic dependence of Tc on holedoping or hole-doping distribution among inequivalent layers, and from an``intrinsic'' contribution. After reviewing our work on the interplay among Tc,hole content, and pressure in the bilayered and multilayered cupratesuperconductors, we will discuss how the proximity to an electronic topologicaltransition (ETT) may give a microscopic justification of the ``intrinsic''pressure dependence of Tc in the cuprates. As a function of the proximity to anETT, we recover a nonmonotonic behaviour of the superconducting gap at T=0,regardless of the pairing symmetry of the order parameter. This is in agreementwith the trend observed for Tc as a function of pressure and other materialspecific quantities in several high-Tc cuprates. In the case of epitaxiallystrained cuprate thin films, we argue that an ETT can be driven by astrain-induced modification of the in-plane band structure, at constant holecontent, at variance with a doping-induced ETT, as is usually assumed. We alsofind that an increase of the in-plane anisotropy enhances the effect offluctuations above Tc on the normal-state transport properties, which is afingerprint of quantum criticality at T=0.

Pressure-induced electronic topological transitions in low dimensional superconductors