${\mathbb{Z}}$, and hence we wondered if HNN extensions could provide new classes of quasi-lattice ordered groups. Spielberg also showed that a groupoid associated to the Baumslag-Solitar semigroup is amenable [@Spielberg2012 Theorem 3.22].

Second, Clark, an Huef and Raeburn examined the phase-transitions of the Toeplitz algebra of the Baumslag-Solitar group [@HuefClarkRaeburn2015]. As part of their investigation they proved that the Baumslag-Solitar group is amenable as a quasi-lattice ordered group. The standard way to prove amenability, introduced by Laca and Raeburn [@LacaRaeburn1996], is to use a “controlled map”: an order-preserving homomorphism between quasi-lattice ordered groups. They observed that the height map, which counts the number of times the stable letter of the HNN extension appears in a word, is almost a controlled map, and then they adapted the standard proof in [@HuefClarkRaeburn2015 Appendix A] to fit.

Our innovation in this paper is a more general definition of a controlled map. We prove in Theorem \[p-controlled map implies amenable\] that if $(G,P)$ is a quasi-lattice ordered group and there is a controlled map $\mu$ into an amenable group, and if $\ker\mu$ is an amenable quasi-lattice ordered group, then $G$ is amenable. The motivation for Theorem \[p-controlled map implies amenable\] was two-fold. First, if 