Appendix: Finite prolongation in RGD-systems (Q2016987)

The present article is an appendix to [\textit{T. Hartnick} et al., ibid. 13, 1--71 (2013; Zbl 1295.51017)], and its primary purpose is to establish a result which is needed there. To state it (or rather, an immediate corollary), we fix some terminology. Let \(\Phi\) be a root system of type \((W,S)\), \(\Pi\subset\Phi\) a set of simple roots and \(\Phi_\pm\) the corresponding sets of positive / negative roots. Let \(G\) be a group with root group datum \((\{U_{\alpha}\}_{\alpha\in\Phi},T)\), and for \(\epsilon\in\{+,-\}\) define \(U_\epsilon:=\langle U_\alpha\mid \alpha\in\Phi_\epsilon \rangle\), \(B_\epsilon:= U_\epsilon T\), \(X_\epsilon:= \bigcup_{\alpha\in\Pi} U_{\epsilon \alpha}\), \(X:=X_+\cup X_-\). (Examples for \(G\) include split Kac-Moody groups, where \(B_+\) and \(B_-\) are Borel subgroups of plus and minus type.) Theorem. Suppose that \((W,S)\) is two-spherical (i.e.\ its Dynkin diagram contains no \(\infty\)-edges), and suppose that within each rank 2 residue \(R\) of the building \(G/B_+\), the chamber system opposite to some chamber \(c\in R\) is connected with finite diameter. Then there is a constant \(k\in\mathbb{N}\) such that for \(f:\mathbb{N}\to\mathbb{N}, n\mapsto nk^n\), the following holds: \[ B_+ B_- \cap X^n \subseteq (U_+ \cap X_+^{f(n)}) (T \cap X_+^{f(n)}X^nX_-^{f(n)}) (U_- \cap X_-^{f(n)}). \]