Topologies on pseudo-trees and applications

“…The dendrons are monotonically normal (see, for example, the proof of Corollary 3.14 in [31]). The dendrons are monotonically normal (see, for example, the proof of Corollary 3.14 in [31]).…”

Section: • ¬(I) ⇒ ¬(Ii)mentioning

confidence: 98%

“…P. de la Harpe and J.-P. Préaux, following N. Monod and Y. Shalom, described a version of the same topology for the case of the space that is the union of a tree and its ends, and referred to it as to "the shadow topology". In the monograph [31] by J. Nikiel, a related topology on pseudotrees played a central role, but also was not named and was denoted by T ≤ . T. Coulbois, A. Hilion, and M. Lustig [12] described this topology for the case of R-trees, calling it "the observers' topology" and attributing the invention of this term to V. Guirardel.…”

Section: Introductionmentioning

confidence: 99%

Pretrees and the shadow topology

Malyutin

2015

St. Petersburg Math. J.

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A further development of the theory of pretrees, started by the work of L. Ward, P. Duchet, B. Bowditch, S. Adeleke and P. Neumann, and others, is presented. In particular, a relationship between this theory and the theory of convex structures is established. The shadow topology is investigated in detail. This remarkable topology emerges on tree-like objects of various types and has broad application.2010 Mathematics Subject Classification. Primary 54F50.

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“…(2) Xa has a least element xa . Theorem 3.1 is a particular case of [N,Theorem 7.6,p. 70]; in the notation of [N], our Xa is the same as C7(([0, 00), <), a).…”

Section: Universal R-tree Of Order Amentioning

confidence: 99%

“…70]; in the notation of [N], our Xa is the same as C7(([0, 00), <), a). General versions of Theorems 3.2 and 3.3 below can be found in [N,Theorem 7.7,p. 72].…”

Section: Universal R-tree Of Order Amentioning

confidence: 99%

Universal spaces for 𝑅-trees

Mayer¹,

Nikiel²,

Oversteegen³

1992

Trans. Amer. Math. Soc.

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Abstract.R-trees arise naturally in the study of groups of isometries of hyperbolic space. An R-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an R-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize R-trees among metric spaces. A universal R-tree would be of interest in attempting to classify the actions of groups of isometries on R-trees. It is easy to see that there is no universal R-tree. However, we show that there is a universal separable R-tree 7n0 . Moreover, for each cardinal a, 3 < a < H0 , there is a space Ta C 7n0 , universal for separable R-trees, whose order of ramification is at most a . We construct a universal smooth dendroid D such that each separable R-tree embeds in D ; thus, has a smooth dendroid compactification. For nonseparable R-trees, we show that there is an R-tree Xa , such that each R-tree of order of ramification at most a embeds isometrically into Xa . We also show that each R-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of R-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.

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Topologies on pseudo-trees and applications

Cited by 14 publications

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Explicit Constructions of Universal ℝ-Trees and Asymptotic Geometry of Hyperbolic Spaces

Explicit Constructions of Universal ℝ-Trees and Asymptotic Geometry of Hyperbolic Spaces

Pretrees and the shadow topology

Universal spaces for 𝑅-trees

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