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A metric between quasi-isometric trees

Author: Álvaro Martínez-Pérez
Journal: Proc. Amer. Math. Soc. 140 (2012), 325-335
MSC (2010): Primary 54E40, 30C65, 53C23; Secondary 54E40
Published electronically: August 11, 2011
MathSciNet review: 2833543
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Abstract: It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete $ \mathbb{R}$-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that when the ultrametric spaces are the corresponding end spaces, this map defines a metric between rooted geodesically complete simplicial trees with minimal vertex degree 3 in the same quasi-isometry class. Moreover, this metric measures how far the trees are from being rooted isometric.

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Additional Information

Álvaro Martínez-Pérez
Affiliation: Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid 28040, Spain

Keywords: Tree, real tree, ultrametric, end space, bounded distortion equivalence, quasi-isometry, PQ-symmetric, pseudo-doubling metric space
Received by editor(s): August 6, 2010
Published electronically: August 11, 2011
Additional Notes: The author was partially supported by MTM 2009-07030.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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