  It is shown that the empirical relations between transition temperature,
normal state conductivity linearly extrapolated to the value at the transition
temperature, zero temperature penetration depths, etc., as observed in a rich
variety of cuprate superconductors, are remarkably consistent with the
universal critical properties of anisotropic systems which fall into the 3D-XY
universality class and undergo a crossover to a quantum critical point in 2D.
The variety includes n- and p-type cuprates, comprises the underdoped and
overdoped regimes and the consistency extends up to six decades in the scaling
variables. The resulting scaling relations for the oxygen isotope and
hydrostatic pressure effects agree with the experimental data and reveal that
these effects originate from local lattice distortions preserving the volume of
the unit cell. These observations single out 3D and anisotropic microscopic
models which incorporate local lattice distortions, fall in the experimentally
accessible regime into the 3D-XY universality class, and incorporate the
crossover to 2D quantum criticality where superconductivity disappears.
