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

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