Boundaries of nonpositively curved groups of the form G × ℤn

Bowers, Philip L.; Ruane, Kim

doi:10.1017/s0017089500031414

Cited by 26 publications

(29 citation statements)

References 7 publications

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“…As proved in [3], this extends to a G v -equivariant homeomorphism ∂X v → ∂X v taking poles to poles and longitudes to longitudes (in fact, this is an isometry in the Tits metric). Given a longitude of ∂X v , we will refer to its image under this homeomorphism as the corresponding longitude of ∂X v .…”

Section: Implications Of Hyperbolicitymentioning

confidence: 75%

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Guilbault

Mooney

2014

Journal of Topology

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Abstract. Question 2.6 of Bestvina's Questions in Geometric Group Theory asks whether every pair of boundaries of a given CAT(0) group G is cell-like equivalent [1]. The question was posed by Bestvina shortly after the discovery, by Croke and Kleiner [5], of a CAT(0) group Γ that admits multiple boundaries. Previously, it had been observed by Bestvina and Geoghegan that all boundaries of a torsion free CAT(0) G would necessarily have the same shape. Since "cell-like equivalence" is weaker than topological equivalence, but in most circumstances, stronger (and more intuitive) than shape equivalence, this question is a natural one when working with the pathological types of spaces that occur as group boundaries. Furthermore, the definition of cell-like equivalence allows for a obvious G-equivariant extension. In private conversations, Bestvina has indicated a preference for the G-equivariant formulation of Q2.6.In this paper we provide a positive answer to Bestvina's G-equivariant Cell-like Equivalence Question for the class of admissible groups studied by Croke and Kleiner in [5]. Since that collection includes the original Croke-Kleiner group Γ, our result provides a strong solution to Q2.6, for the group that originally motivated the question.In [8], we described a general strategy for attacking the following:Bestvina's Equivariant Cell-like Equivalence Question. For a CAT(0) group G, are all boundaries G-equivariantly cell-like equivalent?This question is a strong version of Q2.6 from Bestvina's Questions in Geometric Group Theory [1], where G-equivariance is not required. Both versions of the question were motivated by a desire to understand the now-famous example, due to Croke and Kleiner [5], of a CAT(0) group Γ that admits boundaries that are not topologically equivalent. Prior to the emergence of that example, it had been observed by both Bestvina and Geoghegan that all boundaries of a given CAT(0) group G would necessarily be shape equivalent. The point then is that, in the proper context, cell-like equivalence is a relationship more flexible than topological equivalence, but stronger than mere shape equivalence. In addition, by the nature of its definition, a celllike equivalence allows for a more direct comparison of the spaces involved. One of those ways is that an equivariance requirement can easily be added-a requirement that is obviously desireable in the context of group boundaries. (Definitions of all terms used in this paragraph will be discussed shortly.)In [6], Croke and Kleiner built upon their work in [5] by analyzing a collection of CAT(0) groups which they called admissible, and which we term Croke-Kleiner admissible (or CKA groups). The main result of this paper is the following.Theorem A. If G is a Croke-Kleiner admissible group, then all CAT(0) boundaries of G are G-equivariantly cell-like equivalent.Since the the original Croke-Kleiner group Γ is a CKA group, our most striking application is the corresponding:Corollary. All boundaries of the Croke-Kleiner group Γ are Γ-equivariantly cel...

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Section: Implications Of Hyperbolicitymentioning

confidence: 75%

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Guilbault

Mooney

2014

Journal of Topology

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“…Then we denote the boundary of the rigid CAT(0) group by ∂ . A conclusion in [3, Theorem II.7.1] and [2] implies that if is a rigid CAT(0) group, then × Z n is also a rigid CAT(0) group for each n ∈ N. In [11], K. Ruane has proved that if 1 × 2 is a CAT(0) group and if 1 and 2 are hyperbolic groups (in the sense of Gromov) then 1 × 2 is rigid. On rigidity of products of rigid CAT(0) groups, we obtain the following theorem from Theorem 3 as a generalization of these results.…”

Section: Theoremmentioning

confidence: 99%

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

Hosaka

2011

Math. Z.

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In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group is said to be rigid, if determines its boundary up to homeomorphisms of a CAT(0) space on which acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if 1 and 2 are rigid CAT(0) groups then so is 1 × 2 .

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“…In [7], P. L. Bowers and K. Ruane have constructed an example that the natural quasi-isometry Gx 0 → Gy 0 (gx 0 → gy 0 ) does not extend continuously to any map between the boundaries ∂X and ∂Y of X and Y . Also S. Yamagata [52] has constructed a similar example using a right-angled Coxeter group and its Davis complex.…”

Section: Introductionmentioning

confidence: 99%

“…Now we recall the example of Bowers and Ruane in [7]. Let G = F 2 × Z and X = Y = T × R, where F 2 is the rank 2 free group generated by {a, b} and T is the Cayley graph of F 2 with respect to the generating set {a, b}.…”

Section: Introductionmentioning

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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Boundaries of nonpositively curved groups of the form G × ℤⁿ

Cited by 26 publications

References 7 publications

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence

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

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

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