Scaling entropy growth gap

The scaling entropy of a p.m.p. action is a slow-entropy type invariant that characterizes the intermediate growth of entropy in a dynamical system. An amenable group $G$ has a scaling entropy growth gap if the scaling entropy of any its free p.m.p. action admits a non-trivial lower bound. We prove that the group $S_\infty$ of all finite permutations has a scaling entropy growth gap as well as the Houghton group $\mathcal{H}_2$. By proving this, we show that there are finitely generated amenable groups with this property.