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 ≥ 7 and Stallings’ non-FP₃ group do not satisfy MAC. As a consequence, the condition MAC is not a commensurability invariant.