Polygons in Buildings and their Refined Side Lengths

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber Δeuc. In addition to the metric length it contains information on the direction of the segment. In this paper we study restrictions on the Δeuc-valued side lengths of polygons in Euclidean buildings. The main result is that for thick Euclidean buildings X the set $${\mathcal{P}n(X)}$$ of possible Δeuc-valued side lengths of oriented n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of Δeuc-valued weights of semistable weighted configurations on the Tits boundary ∂Tits X. The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM2]