Automorphisms of non-spherical buildings have unbounded displacement

Abstract

A well-known folklore result says that a nontrivial automorphism φ of a thick Euclidean building X has unbounded displacement. Here we are thinking of X as a metric space, and the assertion is that there is no bound on the distance that φmoves a point. [For the proof, consider the action of φ on the boundaryX∞ at infinity. If φ had bounded displacement, then φwould act as the identity onX∞, and one would easily conclude that φ = id.] In this note we generalize this result to buildings that are not necessarily Euclidean. We work with buildings∆ as combinatorial objects, whose set C of chambers has a discrete metric (“gallery distance”). We say that ∆ is of purely infinite type if every irreducible factor of its Weyl group is infinite.