Small entropy doubling H(R_{2n}) ≤ H(R_n) + log K for a finitely supported symmetric random walk implies the group is virtually nilpotent.
Symmetric random walks on groups.Trans
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Small entropy doubling for random walks and polynomial growth
Small entropy doubling H(R_{2n}) ≤ H(R_n) + log K for a finitely supported symmetric random walk implies the group is virtually nilpotent.