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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 either 0, 1 or 2.