Uniformly continuous maps between ends of $${\mathbb{R}}$$ -trees

Martínez-Pérez, Álvaro; Morón, Manuel A.

doi:10.1007/s00209-008-0431-5

Cited by 22 publications

(16 citation statements)

References 12 publications

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“…For more background on R-trees, see Bestvina [6], Chiswell [13], and Morgan and Shalen [35]. For more information and references on end spaces of R-trees, see Hughes [26] and Martínez-Pérez and Morón [33].…”

Section: Recollections On Trees and Their Endsmentioning

confidence: 99%

Trees, Ultrametrics, and Noncommutative Geometry

Hughes¹

2012

Pure and Applied Mathematics Quarterly

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Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry types and the local similarity types in those spaces can be analyzed using groupoid C * -algebras. The concept of a locally rigid action of a countable group Γ on a compact, ultrametric space by local similarities is introduced. It is proved that there is a faithful unitary representation of Γ into the germ groupoid C * -algebra of the action. The prototypical example is the standard action of Thompson's group V on the ultrametric Cantor set. In this case, the C * -algebra is the Cuntz algebra O 2 and representations originally due to Birget and Nekrashevych are recovered. End spaces of trees are sources of ultrametric spaces. Some connections are made between locally rigid, ultrametric spaces and a concept in the theory of tree lattices of Bass and Lubotzky.

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Section: Recollections On Trees and Their Endsmentioning

confidence: 99%

Trees, Ultrametrics, and Noncommutative Geometry

Hughes¹

2012

Pure and Applied Mathematics Quarterly

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“…By choosing a root on an R-tree the boundary at infinity naturally becomes a complete ultrametric space. In fact, several categorical equivalences are proved in the literature, see [10] and [15]. Further equivalences may be found in [11].…”

Section: Hyperbolic Approximationmentioning

confidence: 96%

Real valued functions and metric spaces quasi-isometric to trees

Martínez-Pérez

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. We prove that if X is a complete geodesic metric space with uniformly generated first homology group and f : X → R is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent condition for a metric space to be PQ-symmetric to an ultrametric space.

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“…If d(v 0 , z) ≤ d(v 1 , z) the vertices u 0 , u 1 hold that d(u 0 , v 0 ) ≤ d(u 0 , v 1 ) and d(u 1 , v 0 ) ≤ d(u 1 , v 1 ) but this last inequality is not possible because there is a geodesic segment from x to u 1 containing v 1 (restriction of γ 1 ) and since Lemma 5. 16. Let x ∈ A ε , {y 1 , · · · , y n } = ∂B(x, ε) ∩ cl(X\B(x, ε)) the partition defined in Lemma 5.13.…”

Section: Finite Graphsmentioning

confidence: 99%

Semiflows induced by length metrics: On the way to extinction

Martínez-Pérez

Morón

2016

Topology and its Applications

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Bing and Moise proved, independently, that any Peano continuum admits a length metric d. We treat non-degenerate Peano continua with a length metric as evolution systems instead of stationary objects. For any compact length space (X, d) we consider a semiflow in the hyperspace 2 X of all non-empty closed sets in X. This semiflow starts with a canonical copy of the Peano continuum (X, d) at t = 0 and, at some time, collapses everything into a point. We study some properties of this semiflow for several classes of spaces, manifolds, graphs and finite polyhedra among them.

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Uniformly continuous maps between ends of $${\mathbb{R}}$$ -trees

Cited by 22 publications

References 12 publications

Trees, Ultrametrics, and Noncommutative Geometry

Trees, Ultrametrics, and Noncommutative Geometry

Real valued functions and metric spaces quasi-isometric to trees

Semiflows induced by length metrics: On the way to extinction

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