Which physical symmetries can be imposed by averaging?

Averaging a trained model over a compact group makes it exactly equivariant, never increases its risk on an invariant data distribution, and is the averaging operation that data augmentation performs. The symmetry groups of physical law are not compact. We partition the non-compact case by two criteria that come apart. Ball-truncated averaging is a Følner average exactly when the group has subexponential volume growth; under exponential growth rate $p$ the outer unit shell carries a fraction $1-e^{-p}$ of the mass at every cutoff, and the average converges to an average over the maximal compact subgroup at the cutoff, not to a single group element. Non-compactness alone voids the risk guarantee: we give an exact counterexample on translations, a polynomial-growth group, and measure averaging raising risk in 33% of instances under exponential growth and 28% under polynomial growth against 0% for a compact group. Amenability decides neither criterion but decides whether a repair exists, and we give the Følner sequences for $\mathrm{Aff}(1)$ and $\mathrm{SIM}(2)$ whose balls are not one.