Small entropy doubling for random walks and polynomial growth

TL;DR

This paper explores entropy analogues of small doubling properties in random walks on groups, establishing criteria for polynomial growth and virtually nilpotent groups based on entropy behavior.

Contribution

It introduces entropy-based criteria for polynomial growth and virtually nilpotent groups, adapting Tao's entropy methods to broader group settings.

Findings

If entropy doubles slowly, the group is virtually nilpotent.

Entropy growth is superlogarithmic for non-virtually nilpotent groups.

Provides entropy criteria for polynomial growth in groups.

Abstract

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.