Twin trees and 𝜆_{Λ}-gons

Abstract

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