---
abstract: 'The Baumslag-Solitar group is an example of an HNN extension. Spielberg showed that it has a natural positive cone, and that it is then a quasi-lattice ordered group in the sense of Nica. We give conditions for an HNN extension of a quasi-lattice ordered group $(G,P)$ to be quasi-lattice ordered. In that case, if $(G,P)$ is amenable as a quasi-lattice ordered group, then so is the HNN extension.'
address: 'Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand.'
author:
- Astrid an Huef
- Iain Raeburn
- Ilija Tolich
date: '8 March, 2017'
title: 'HNN extensions of quasi-lattice ordered groups and their operator algebras'
---

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Introduction
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Since they were introduced by Nica [@Nica1992], quasi-lattice ordered groups and their $C^*$-algebras have generated considerable interest (see, for example, [@LacaRaeburn1996],[@LacaRaeburn2010]). The amenability of quasi-lattice ordered groups has been a deep subject (see, for example, [@CrispLaca2002],[@CrispLaca2007] and [@Li2012]). Quasi-lattice ordered groups are also examples of the more recent LCM semigroups [@BrownloweLarsenStammeier2016], [@Starling2015]. Here we generalise two recent results about the Baumslag-Solitar group.

First, Spielberg proved that the Baumslag-Solitar group is quasi-lattice ordered [@Spielberg2012]. The Baumslag-Solitar group is an example of an HNN extension of 