\leq \mu_G(A_1+B) + \mu_G(A_2+B),$$ valid for any compact $A_1,A_2,B \subset G$, (which follow from the identity $(A_1 \cup A_2)+B = (A_1+B) \cup (A_2+B)$ and the inclusion $(A_1 \cap A_2)+B \subset (A_1+B) \cap (A_2+B)$ respectively), one can obtain a number of closure properties regarding critical pairs, for instance establishing that if $(A_1,B)$ and $(A_2,B)$ are critical pairs then $(A_1 \cup A_2,B)$ and $(A_1 \cap A_2,B)$ are also, provided that $A_1 \cap A_2$ is non-empty and $A_1 \cup A_2$ is not too large. Similarly, using the associativity $(A+B)+C = A+(B+C)$ of the sum set operation, one can show that if $(A,B)$ and $(A+B,C)$ are critical pairs, then so are $(B,C)$ and $(A,B+C)$. Using such closure properties repeatedly in combination with the translation invariance of the critical pair concept, we can start with a critical pair $(A,B)$ and generate a small (but non-trivial) auxiliary set $C$ such that $(A,C)$ and $(C,C)$ are critical pairs; furthermore, we can also arrange matters so that $(C,kC)$ is a critical pair for all bounded $k$ (e.g. all $1 \leq k \leq 10^4$), where $kC = C + \dots + C$ is the $k$-fold iterated sumset of $C$. This implies in particular that $C$ has linear growth in 