Critical constants for recurrence on groups of polynomial growth

The critical constant for recurrence, c(rt), is an invariant of the quotient space H\G of a finitely generated group. The constant is determined by the largest moment a probability measure on G can have without the induced random walk on H\G being recurrent. We present a description of which subgroups of groups of polynomial volume growth are recurrent. Using this we show that for such recurrent subgroups c(rt) corresponds to the relative growth rate of H in G, and in particular c(rt) is an element of {0, 1, 2}.