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

2005/10/24 by Tetsuya Hosaka, Hosaka, Tetsuya · 1 citation
Mathematics · #57M07 #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.math/0510509

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

Abstract

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 F_g be the fixed-point set of g in the boundary ∂ X. Then we show that F_g=L(Z_g), where Z_g is the centralizer of g (i.e. Z_g=\v∈ G| gv=vg\) and L(Z_g) is the limit set of Z_g in ∂ X. Thus we obtain that F_g≠ ∅ if and only if the set Z_g is infinite. We also show that if g is a hyperbolic isometry, then F_g=∂\Min(g), where ∂\Min(g) is the boundary of the minimal set \Min(g) of g. This implies that the fixed-point set F_g and the periodic-point set P_g of g in ∂ X have suspension forms.

Cited by

• Boundary actions of CAT(0) spaces: topological freeness and applications to C^*-algebras

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