Small entropy doubling for random walks and polynomial growth

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property for random walks on groups. We show that if a finitely supported symmetric random walk $R_n$ satisfies $$\mathrm{H}(R_{2n}) \le \mathrm{H}(R_n) + \log K$$ at some sufficiently large scale $n$, then the underlying group is virtually nilpotent, with bounds depending on $K$ and $\mu_{\min}$. Our approach adapts Tao's entropy Balog--Szemer\'edi--Gowers argument to unimodular locally compact groups, combined with structural results on approximate groups. As applications, we obtain entropy-based criteria for polynomial growth. We also deduce an entropy gap phenomenon: if $G$ is not virtually nilpotent, then the entropy of random walks on $G$ grows faster than a universal superlogarithmic function.