The geometry of the chamber system of a semimodular lattice

In this paper geometric properties of the following metric space C are studied. Its elements are called chambers and are the maximal chains of a semimodular lattice X of finite height and its metric d is the gallery distance. We show that X has many properties in common with buildings. More specifically, Tits 17] has recently described buildings in terms of Weyl-group valued distance functions. We consider the Jordan-Hölder permutation (C, D) corresponding to a pair C, D of chambers and show that it has most properties of such a distance with values in the symmetric group.