Extremal elements in Coxeter groups and metric commensurators of Kac-Moody groups (Q1706090)

Let \((W,S)\) be a Coxeter system. For two elements \(w, v \in W\), the authors write \(w\vartriangleleft v\) if there exists an \(s \in S\) such that \(l(ws) = l(w) + 1\) and \(v = sws\). Hence, \(w\vartriangleleft v\) implies \(l(w) \leq l(v)\). The authors call a non-identity element \(w\) extremal in \((W, S)\) if \(\ell(w) = \ell(v)\) for every sequence \(w=w_0 \vartriangleleft w_1 \vartriangleleft \cdots \vartriangleleft w_k = v\) in \(W\). Two twinnings between two buildings with chamber sets \(\mathcal{C}_+\) and \(\mathcal{C}_-\) are said to be at finite distance if their sets of opposite chamber pairs are at finite Hausdorff distance from each other in \(\mathcal{C}_+ \times \mathcal{C}_-\). The main results that the authors show are the following theorems. Theorem 1. Let \((W, S)\) be an irreducible Coxeter system such that there exists an extremal element \(w \in W\). Then \((W, S)\) is either spherical or affine. Theorem 2. Let \(\Delta_+, \Delta_-\) be two buildings of irreducible type \((W, S)\). If there exist two distinct twinnings at finite distance between them, then there exists an extremal element in \(W\). In particular, \((W, S)\) is spherical or affine. Theorem 3. The metric commensurator of an irreducible Kac-Moody group acting on a product of two non-affine buildings coincides with its normalizer.