Minimal Almost Convexity

TL;DR

This paper investigates the minimal almost convexity property in certain groups, demonstrating that some groups are minimally almost convex while others are not, and analyzing the implications for group invariants.

Contribution

It establishes that $BS(1,2)$ is minimally almost convex and compares this property with Poénaru's condition, showing differences and non-invariance under commensurability.

Findings

$BS(1,2)$ is minimally almost convex.

$BS(1,2)$ does not satisfy Poénaru's $P(2)$.

Groups $BS(1,q)$ for $q \,\geq\, 7$ and Stallings' group are not minimally almost convex.

Abstract

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 Po\'enaru'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.