Twin trees and $\lambda _{\Lambda }$-gons

We define a natural generalization of generalized $n$-gons to the case of $\Lambda$-graphs (where $\Lambda$ is a totally ordered abelian group and $0<\lambda \in \Lambda$). We term these objects $\lambda _{\Lambda }$-gons. We then show that twin trees as defined by Ronan and Tits can be viewed as $(1,0)_{\Lambda }$-gons, where $\Lambda = Z \times Z$ is ordered lexicographically. This allows us to then generalize twin trees to the case of $\Lambda$-trees. Finally, we give a free construction of $\lambda _{\Lambda }$-gons in the cases where $\Lambda$ is discrete and has a subgroup of index $2$ that does not contain the minimal element of $\Lambda$.