  We strengthen the maximal ergodic theorem for actions of groups of polynomial
growth to a form involving jump quantity, which is the sharpest result among
the family of variational or maximal ergodic theorems. As a consequence, we
deduce in this setting the quantitative ergodic theorem, in particular, the
upcrossing inequalities with exponential decay. The ideas or techniques involve
probability theory, non-doubling Calder\'on-Zygmund theory, almost
orthogonality argument and some delicate geometric argument involving the balls
and the cubes on the group equipped with a not necessarily doubling measure.
