Rank-one isometries of CAT(0) cube complexes and their centralisers

Abstract

Abstract If G is a group acting geometrically on a CAT(0) cube complex X and if g ∈ G has infinite order, we show that exactly one of the following situations occurs: (i) g defines a rank-one isometry of X; (ii) the stable centraliser SCG (g) = {h ∈ G ∣ ∃ n ≥ 1, [h, gn ] = 1} of g is not virtually cyclic; (iii) Fix Y (gn ) is finite for every n ≥ 1 and the sequence (Fix Y (gn )) takes infinitely many values, where Y is a cubical component of the Roller boundary of X which contains an endpoint of an axis of g. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.