A metric between quasi-isometric trees

Martínez-Pérez, Álvaro

doi:10.1090/S0002-9939-2011-11286-3

A metric between quasi-isometric trees
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by Álvaro Martínez-Pérez PDF

Proc. Amer. Math. Soc. 140 (2012), 325-335 Request permission

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.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266–306. MR 1771428, DOI 10.1007/s000390050009
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Sergei Buyalo and Viktor Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. MR 2327160, DOI 10.4171/036
V. Z. Feĭnberg, Compact ultrametric spaces, Dokl. Akad. Nauk SSSR 214 (1974), 1041–1044 (Russian). MR 0348715
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
Bruce Hughes, Trees and ultrametric spaces: a categorical equivalence, Adv. Math. 189 (2004), no. 1, 148–191. MR 2093482, DOI 10.1016/j.aim.2003.11.008
Bruce Hughes, Álvaro Martínez-Pérez, and Manuel A. Morón, Bounded distortion homeomorphisms on ultrametric spaces, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 473–492. MR 2731702, DOI 10.5186/aasfm.2010.3529
Álvaro Martínez-Pérez and Manuel A. Morón, Uniformly continuous maps between ends of $\Bbb R$-trees, Math. Z. 263 (2009), no. 3, 583–606. MR 2545858, DOI 10.1007/s00209-008-0431-5
Á. Martínez-Pérez, Quasi-isometries between visual hyperbolic spaces. Manuscripta Math. (2011), DOI 10.1007/s00229-011-0463-8.
Lee Mosher, Michah Sageev, and Kevin Whyte, Quasi-actions on trees. I. Bounded valence, Ann. of Math. (2) 158 (2003), no. 1, 115–164. MR 1998479, DOI 10.4007/annals.2003.158.115
Frédéric Paulin, Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. (2) 54 (1996), no. 1, 50–74 (French, with French summary). MR 1395067, DOI 10.1112/jlms/54.1.50