Structures of sets with minimal measure growth in connected unimodular groups

Let $G$ be a connected unimodular group equipped with a (left and hence right) Haar measure $\mu_G$, and suppose $A, B \subseteq G$ are nonempty and compact. An inequality by Kemperman gives us $$\mu_G(AB)\geq\min\{\mu_G(A)+\mu_G(B),\mu_G(G)\}.$$ This implies that the $n$-fold product $A^n$ of $A$ has $\mu_G( A^n) \geq \min\{n\mu_G(A),\mu_G(G)\}$ and when $G$ is not compact, $\lim_{n \to \infty} \mu_G( A^n)/ (n \mu_G(A)) \geq 1$. We obtain simple classifications of $G$, $A$, and $B$ such that the equalities hold, answering a question asked by Kemperman in 1964. We also get near equality versions of the above results with explicit bound for compact $G$, confirming conjectures made by Griesmer and by Tao. As applications, we prove measure expansion gap results for connected compact simple Lie groups, and obtained an improved bound for the inverse theorem of Kneser's inequality. Our strategy of proof involves reducing the problem to Lie groups through the use of an argument inspired by Breuillard-Green-Tao's classification of approximate groups via Hrushovski's Lie model theorem, and then developing new techniques to reduce the Lie group problem to the abelian case.