Let $S$ be a symmetric subset of a group $G$ containing the identity, and let $S^n$ be the set of all products of $n$ elements of $S$. If $S^3\subset gS$ for some translate $gS$ of $S$ then it follows that $S^2=S^3$. My question is, if $S^2\subset gS$ does it follow that $S=S^2$?
My question is about the growth of a group in geometric group theory. For a finitely generated group $G$ the growth $G$ is the function that assigns to each natural number $n$ the cardinality $|S^n|$ for some finite generating set $S$. Up to an equivalence of growth functions this does not depend on the generating set chosen. Further, the function $(S^n:S)$ that assigns to each $n$ the smallest number of translates of $S$ that cover $S^n$ is in the equivalence class of $|S^n|$. So my question is asking in a general group $G$ does $(S^2:S)=1$ imply $S=S^2$. For $G$ finitely generated by $S$ the answer is yes but trivially because $gS$ and $S$ have the same cardinality.
In general I'm interested in the situation where $G$ is a topological group equipped with a coarse structure and $S$ is a small subset that generates $G$. So for example, it is relevant to assume $G$ is connected and Haar measurable and $S$ is a precompact open set.
Thank you for considering my question.