On fixed-point sets in the boundary of a CAT(0) space

Original · EN

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≠ if and only if the set Zg is infinite. We also show that if g is a hyperbolic isometry, then Fg=∂(g), where ∂(g) is the boundary of the minimal set (g) of g. This implies that the fixed-point set Fg and the periodic-point set Pg of g in ∂ X have suspension forms.