On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

Hosaka, Tetsuya

doi:10.1007/s00209-011-0972-x

Cited by 12 publications

(14 citation statements)

References 13 publications

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“…By splitting theorems (cf. [22], [27]), we obtain (10) ⇒ ( 12) and ( 11) ⇒ (12) (cf. [20,Proposition 6.3]).…”

Section: Rank-one Isometries Of Coxeter Groups and Topological Fracta...mentioning

confidence: 69%

On boundaries of Coxeter groups and topological fractal structures

Hosaka¹

2009

Preprint

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In this paper, based on research on rank-one isometries by W. Ballmann and M. Brin and recent research on rankone isometries of Coxeter groups by P. Caprace and K. Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter group and introduce some observations on boundaries of CAT(0) groups with rank-one isometries.

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“…By splitting theorems (cf. [22], [27]), we obtain (10) ⇒ ( 12) and ( 11) ⇒ (12) (cf. [20,Proposition 6.3]).…”

Section: Rank-one Isometries Of Coxeter Groups and Topological Fracta...mentioning

confidence: 69%

On boundaries of Coxeter groups and topological fractal structures

Hosaka¹

2009

Preprint

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“…Here we note that if W = F × G acts geometrically on a CAT(0) space X then the boundary ∂X is homeomorphic to ∂F * ∂Y where Y is some CAT(0) space on which G acts geometrically by the splitting theorem in [35]. Hence W is rigid if and only if G is rigid.…”

Section: On Equivariant Rigid As Reflection Groupsmentioning

confidence: 99%

“…Here we can find some recent research on CAT(0) groups and their boundaries in [11], [13], [22], [27], [35], [36], [39], [41], [43], [44], [48] and [51]. Details of Coxeter groups and Coxeter systems are found in [6], [9] and [37], and details of Davis complexes which are CAT(0) spaces defined by Coxeter systems and their boundaries are found in [15], [16] and [47].…”

Section: Remarks and Questionsmentioning

confidence: 99%

On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups

Hosaka

2015

Differential Geometry and its Applications

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Abstract. 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 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.

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“…By a splitting theorem for CAT(0) spaces [23,24], we obtain that, if a CAT(0) group G contains a finite-index subgroup G 1 ×G 2 such that G 1 and G 2 are infinite, then a CAT(0) space X on which G acts geometrically contains a quasi-dense subspace which splits as a product X 1 × X 2 , where X 1 and X 2 are unbounded [9,Proposition 6.3]. This implies the following corollary.…”

Section: Corollary 54 Suppose That a Group G Acts Geometrically On mentioning

confidence: 99%

CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

Hosaka

2010

Journal of Pure and Applied Algebra

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Communicated by C.A. Weibel MSC: 20F65 20F55 57M07 a b s t r a c tIn this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a proper CAT(0) space X . (Such a group G is called a CAT(0) group.) Then the group G acts by homeomorphisms on the boundary ∂X of X and we can define a metric d ∂X on the boundary ∂X. The boundary ∂X is called a scrambled set if, for any α, β ∈ ∂X with α = β, (1) lim sup{d ∂X (gα, gβ) | g ∈ G} > 0 and (2) lim inf{d ∂X (gα, gβ) | g ∈ G} = 0. We investigate when boundaries of CAT(0) groups (and Coxeter groups) are scrambled sets.

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On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

Cited by 12 publications

References 13 publications

On boundaries of Coxeter groups and topological fractal structures

On boundaries of Coxeter groups and topological fractal structures

On equivariant homeomorphisms of boundaries of CAT(0) groups and Coxeter groups

CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

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