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].
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Kapovich, M., Leeb, B. & Millson, J.J. Polygons in Buildings and their Refined Side Lengths. Geom. Funct. Anal. 19, 1081–1100 (2009). https://doi.org/10.1007/s00039-009-0026-2
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DOI: https://doi.org/10.1007/s00039-009-0026-2