Polygons in Buildings and their Refined Side Lengths

Abstract

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 Δ euc . In addition to the metric length it contains information on the direction of the segment. In this paper we study restrictions on the Δ euc -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 Δ euc -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 Δ euc -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].