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