Rank-One Isometries of Buildings and Quasi-Morphisms of Kac-Moody Groups

Given an irreducible non-spherical non-affine (possibly non-proper) building X, we give sufficient conditions for a group G < Aut(X) to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies in particular to all irreducible (non-spherical and non-affine) Kac-Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov [MK, Prob. 14.13]. Independently of these considerations, we also include a discussion of rank-one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous