Dense Subsets of Boundaries of Cat(0) Groups

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group G acts geometrically on a CAT(0) space X and suppose that there exists an element g0 ∈ G such that (1) Zg0 is finite, (2) X \ Fg0 is not connnected, and (3) each component of X \ Fg0 is convex and not g0-invariant, where Zg0 is the centralizer of g0 and Fg0 is the fixed-point set of g0 in X (that is, Zg0 = {h ∈ G| g0h = hg0} and Fg0 = {x ∈ X | g0x = x}). Then we show that each orbit Gα is dense in the boundary ∂X (i.e. ∂X is minimal) and the set {g∞ | g ∈ G, o(g) = ∞} is also dense in the boundary ∂X . We obtain an application for dense subsets on the boundary of a Coxeter system.