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Some results on maps that factor through a tree. (English) Zbl 1317.51008
A tree is a metric space $$T$$ that is uniquely arc-connected, i.e., for different points $$p, p' \in T$$ there is an embedding $$\gamma: [0,1]\rightarrow T$$ with $$\gamma(0)=p, \gamma(1)=p'$$ and any other such embedding is a reparameterization of $$\gamma$$.
Let $$\varphi : X \rightarrow Y$$ be a uniformly continuous map between metric spaces. The mapping $$\varphi$$ is said to have Property (T) if for all $$x, x' \in X$$ with $$\varphi (x) \neq \varphi(x')$$ there is a point $$y \in Y \backslash \{ \varphi(x), \varphi(x')\}$$ such that for any curve $$\gamma: [0, 1] \rightarrow X$$ connecting $$x$$ with $$x', y$$ is contained in $$\text{im}(\varphi\circ\gamma)$$.
The author characterizes maps $$\varphi$$ which factor through a tree. The Property (T) is frequently used throughout the paper. Theorem 1.2 gives a characterization of Property (T).

##### MSC:
 51F99 Metric geometry 30L99 Analysis on metric spaces 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 26A42 Integrals of Riemann, Stieltjes and Lebesgue type 55M25 Degree, winding number
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##### References:
  L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80. · Zbl 0984.49025  W. Blaschke, Kreis und Kugel, Veit, Leipzig, 1916.  H. Boedihardjo, H. Ni and Z. Qian, Uniqueness of signature for simple curves, Journal of Functional Analysis 267 (2014), 1778-1806. · Zbl 1294.60063  D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, Volume 33, American Mathematical Society, Providence, 2001. · Zbl 0981.51016  L. Capogna, D. Danielli, S. Pauls and J. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, Volume 259, Birkhäuser, Basel, 2007. · Zbl 1138.53003  H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  P. K. Fritz and M. Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, 2014.  M. Gromov, Carnot-Carathéodory spaces seen from within, in Sub-Riemannian geometry, Progress inMathematics, Volume 144, Birkhäuser, Basel, 1996, 79-323.  M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, Volume 152, Birkhäuser Boston Inc., Boston, 1999. With appendices by M. Katz, P. Pansu and S. Semmes. · Zbl 0953.53002  U. Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683-742. · Zbl 1222.49055  U. Lang, T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 58 (2005), 3625-3655. · Zbl 1095.53033  E. Le Donne and R. Züst, Some properties of Hölder surfaces in the Heisenberg group, Illinois J. Math. 57 (2013), 229-249. · Zbl 1294.53033  T. J. Lyons, Differential equations driven by rough signals, Rev. Math. Iberoamericana. 14 (1998), 215-310. · Zbl 0923.34056  E. Outerelo and J. M. Ruiz, Mapping degree theory, Graduate Studies in Mathematics, Volume 108, American Mathematical Society, Providence, 2009. · Zbl 1183.47056  J. C. Mayer and L. G. Oversteegen, A topological characterization of R-trees, Trans. Amer. Math. Soc. 320 (1990), 395-415. · Zbl 0729.54008  C. Riedweg and D. Schäppi, Singular (Lipschitz) homology and homology of integral currents, arXiv:0902.3831, 2009.  S. Wenger and R. Young, Lipschitz homotopy groups of the Heisenberg groups, Geom. Funct. Anal. 24 (2014), 387-402. · Zbl 1310.57045  L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), 251-282. · JFM 62.0250.02  R. Züst, Currents in snowflaked metric spaces, phd thesis, ETH Zurich, 2011.  R. Züst, Integration of Hölder forms and currents in snowflake spaces, Calc. Var. and PDE 40 (2011), 99-124. · Zbl 1219.49036
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