Polygons in buildings and their refined side lengths (Q2655443)

Let \(S\) be a unit sphere and \(W_{\text{sph}}\) be a finite subgroup of Isom\((S)\) generated by reflections (i.e. an involutive isometry with fixed-point set a great sphere of codimension 1). The pair \((S, W_{\text{sph}})\) is called a spherical Coxeter complex. \(W_{\text{sph}}\) is called the Weyl group and the fixed point sets of the reflections in \(W_{\text{sph}}\) are called walls. The spherical model Weyl chamber, denoted by \(\Delta _{\text{sph}}\), is \(S/W_{\text{sph}}\). An Euclidean Coxeter complex is a pair \((E, W_{\text{aff}})\), where \(E\) is an Euclidean space and \(W_{\text{aff}}\subset \text{Isom}(E)\) is a subgroup generated by reflections (reflections at an affine hyperplane). Let us suppose that the induced reflection group on the sphere at infinity \(\partial _{\text{Tits}}E\) is finite. Take the unit sphere \(S\) to be \(\partial _{\text{Tits}}E\) and \(W_{\text{sph}}:=\text{rot}(W_{\text{aff}})\) where \(\text{rot}:\text{Isom}(E)\to \text{Isom}(\partial_{\text{Tits}}E)\) maps an affine transformation to its linear part, and consider the associated spherical Coxeter complex. The Euclidean model Weyl chamber \(\Delta_{\text{euc}}\) of \((E,W_{\text{aff}})\) is the Euclidean cone over \(\Delta_{\text{sph}}\). Let \((p,q)\) be a pair of points in \(E\). To this pair, one can associate the image in \(\Delta_{\text{euc}}\) of the translation carrying \(p\) to \(q\). This vector \(\sigma (p,q)\) is called the \(\Delta\)-length of the oriented geodesic segment \(\overline{pq}\). In this paper, the authors prove that, for thick Euclidean buildings \(X\), the set \({\mathcal P}_n (X)\) of possible \(\Delta_{\text{euc}}\)-valued side lengths of oriented \(n\)-gons depends only on the associated spherical Coxeter complex and is the space of \(\Delta _{\text{euc}}\)-valued weights of semistable weighted configurations on the Tits boundary \(\partial_{\text{Tits}}X\). This is a consequence of the method used to prove the theorem. The authors use a relation between polygons in Euclidean buildings and weighted configurations on their spherical Tits buildings at infinity via Gauss map type constructions. By a weight configuration on a spherical building \(B\), the authors mean a map \(\psi :( \mathbb {Z} /n\mathbb {Z}, \nu )\rightarrow B\) from a finite measure space. Composing \(\psi\) with the natural projection from \(B\) to \(\Delta_{\text{sph}}\), one obtains a new map \((\mathbb {Z}/n \mathbb {Z}, \nu )\rightarrow \Delta_{\text{sph}}\). The corresponding point in \(\Delta^n_{\text{euc}}\) is called the \(\Delta\)-weights of the configuration \(\psi\).