Cutoff for random walks on nilpotent groups

We study continuous-time random walks on finite nilpotent groups, allowing non-symmetric and nonuniform increment measures. Under a simple degree condition, sharp up to an absolute constant factor, we prove that total variation cutoff occurs if and only if the relaxation time is of smaller order than the mixing time. We establish a quantitative reduction to the Abelianization that preserves relaxation times and compares mixing profiles. Combining this reduction with a Gaussian approximation and the third-named author's cutoff criterion for diffusions yields the cutoff characterization.