On Fixed-point Sets in the Boundary of a Cat(0) Space

In this paper, we investigate the fixed-point set of an element of a CAT(0) group in its boundary. Suppose that a group G acts geometrically on a CAT(0) space X . Let g ∈ G and let Fg be the fixed-point set of g in the boundary ∂X . Then we show that Fg = L(Zg), where Zg is the centralizer of g (i.e. Zg = {v ∈ G| gv = vg}) and L(Zg) is the limit set of Zg in ∂X . Thus we obtain that Fg 6= ∅ if and only if the set Zg is infinite. We also show that if g is a hyperbolic isometry, then Fg = ∂ Min(g), where ∂ Min(g) is the boundary of the minimal set Min(g) of g. This implies that the fixed-point set Fg and the periodic-point set Pg of g in ∂X have suspension forms.