Dense subsets of boundaries of CAT(0) groups

2005/10/24 by Tetsuya Hosaka, Hosaka, Tetsuya
Computer Science · Engineering · Mathematics · #57M07 #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.math/0510511

openalex publication_date 2005/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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) Z_g0 is finite, (2) X∖ F_g0 is not connnected, and (3) each component of X∖ F_g0 is convex and not g₀-invariant, where Z_g0 is the centralizer of g₀ and F_g0 is the fixed-point set of g₀ in X (that is, Z_g0=\h∈ G| g₀h=hg₀\ and F_g0=\x∈ X| g₀x=x\). Then we show that each orbit G α 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.

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