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 ∆euc. In addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the ∆euc-valued side lengths of polygons in Euclidean buildings. The main result is that for thick Euclidean buildings X the set Pn(X) of possible ∆euc-valued side lengths of oriented n-gons, n ≥ 3, depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of ∆euc-valued weights of semistable weighted configurations on the Tits boundary ∂TitsX. 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 [KLM3].