2013/09/10 by Tetsuya Hosaka, Hosaka, Tetsuya
Mathematics · #20F55 #20F65 #57M07 #Advanced Combinatorial Mathematics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.DG #math.GR #math.GT #msc:20F55 #msc:20F65 #msc:57M07
paper · pdf · doi:10.48550/arxiv.1309.2518
31 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1004.4376
openalex publication_date 2013/09/10 · arxiv created 2014/04/03 · arxiv updated 2014/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate an equivariant homeomorphism of the boundaries ∂ X and ∂ Y of two proper CAT(0) spaces X and Y on which a CAT(0) group G acts geometrically. We provide a sufficient condition and an equivalent condition to obtain a G-equivariant homeomorphism of the boundaries ∂ X and ∂ Y as a continuous extension of the quasi-isometry ϕ:Gx0→ Gy0 defined by ϕ(gx0)=gy0, where x0∈ X and y0∈ Y. In this paper, we say that a CAT(0) group G is \it equivariant (boundary) rigid, if G determines its ideal boundary by the equivariant homeomorphisms as above. As an application, we introduce some examples of (non-)equivariant rigid CAT(0) groups and we show that if Coxeter groups W1 and W2 are equivariant rigid as reflection groups, then so is W1 * W2. We also provide a conjecture on non-rigidity of boundaries of some CAT(0) groups.