Abstract: The classical Mean Ergodic Theorem for probability-preserving transformations asserts convergence of a sequence of ergodic averages to some invariant function, but gives no effective control over how fast is this convergence (say, in norm). Truly quantitative versions of this theorem turn out to be quite subtle: one must search not for convergence to a limit function, but for long `epochs' of time over which the ergodic averages are only approximately invariant and remain approximately constant. In this talk I will show a somewhat surprising application of these quantitative results to a problem in geometric group theory, asking for the least possible distortion to the word metric on the discrete Heisenberg group when that group is Lipschitzly embedded as a metric space into a Lebesgue space L^p for some p \in (0,\infty). We obtain essentially sharp bounds on the `compression exponent' which quantifies this distortion of the word metric after reducing to a problem about probability-preserving actions, but only by making essential use of one of the quantitative variants of the ergodic theorem. Based on joint work with Assaf Naor and Romain Tessera.