  In physics, two systems that radically differ at short scales can exhibit
strikingly similar macroscopic behaviour: they are part of the same
long-distance universality class. Here we apply this viewpoint to geometry and
initiate a program of classifying homogeneous metrics on group manifolds by
their long-distance properties. We show that many metrics on low-dimensional
Lie groups have markedly different short-distance properties but nearly
identical distance functions at long distances, and provide evidence that this
phenomenon is even more robust in high dimensions. An application of these
ideas of particular interest to physics and computer science is complexity
geometry--the study of quantum computational complexity using Riemannian
geometry. We argue for the existence of a large universality class of
definitions of quantum complexity, each linearly related to the other, a much
finer-grained equivalence than typically considered. We conjecture that a new
effective metric emerges at larger complexities that describes a broad class of
complexity geometries, insensitive to various choices of microscopic penalty
factors. We discuss the implications for recent conjectures in quantum gravity.
