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Minimal Almost Convexity

In this article we show that the Baumslag-Solitar group $BS(1,2)$ is minimally almost convex, or $MAC$. We also show that $BS(1,2)$ does not satisfy Poénaru's almost convexity condition $P(2)$, and hence the condition $P(2)$ is strictly stronger than $MAC$. Finally, we show that the groups $BS(1,q)$ for $q \geq 7$ and Stallings' non-$FP_3$ group do not satisfy $MAC$. As a consequence, the condition $MAC$ is not a commensurability invariant.