# HNN extensions of quasi-lattice ordered groups and their operator algebras

## TL;DR

This paper investigates conditions under which HNN extensions of quasi-lattice ordered groups remain quasi-lattice ordered and amenable, extending the understanding of their operator algebra structures.

## Contribution

It provides new criteria for HNN extensions of quasi-lattice ordered groups to preserve quasi-lattice order and amenability.

## Key findings

- Conditions for HNN extensions to be quasi-lattice ordered
- HNN extensions inherit amenability from base groups under certain conditions
- Application to Baumslag-Solitar groups and their operator algebras

## 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.