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Suppose $G$ is a connected noncompact locally compact group, $A,B$ arenonempty and compact subsets of $G$, $\mu$ is a left Haar measure on $G$.Assuming that $G$ is unimodular, and $ \mu(A^2) < K \mu(A) $ with $K>1$ a fixedconstant, our first result shows that there is a continuous surjective grouphomomorphism $\chi: G\to L$ with compact kernel, where $L$ is a Lie group with$$\dim(L) \leq \lfloor\log K\rfloor(\lfloor\log K\rfloor+1)/2.$$ We alsodemonstrate that this dimension bound is sharp, establish the relationshipbetween $A$ and its image under the quotient map, and obtain a more generalversion of this result for the product set $AB$ without assuming unimodularity.  Our second result classifies $G,A,B$ where $A,B$ have nearly minimalexpansions (when $G$ is unimodular, this just means $\mu(AB)$ is close to$\mu(A)+\mu(B)$). This answers a question suggested by Griesmer and Tao, andcompletes the last open case of the inverse Kemperman problem.  The proofs of both results involve a new analysis of locally compact group$G$ with bounded $n-h$, where $n-h$ is an invariant of $G$ appearing in therecently developed nonabelian Brunn-Minkowski inequality. We also generalizeRuzsa's distance and related results to possibly nonunimodular locally compactgroups.

On the small measure expansion phenomenon in connected noncompact nonabelian groups