About some properties of similarly homogeneous $\mathbb {R}$-trees

A. I. Bulygin

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 1, pp. 32-42

Résumé

In this paper we consider the properties of locally complete similarly homogeneous inhomogeneous $\mathbb{R}$-trees. The geodesic space is called $\mathbb{R}$-tree if any two points may be connected by the unique arc. The general problem of A. D. Alexandrov on the characterization of metric spaces is considered. The distance one preserving mappings are constructed for some classes of $\mathbb{R}$-trees. To do this, we use the construction with the help of which a new special metric is introduced on an arbitrary metric space. In terms of this new metric, a criterion is formulated that is necessary for a so that a distance one preserving mapping to be isometric. In this case, the characterization by A. D. Alexandrov is not fulfilled. Moreover, the boundary of a strictly vertical $\mathbb{R}$-tree is also studied. It is proved that any horosphere in a strictly vertical $\mathbb{R}$-tree is an ultrametric space. If the branch number of a strictly vertical $\mathbb{R}$-tree is not greater than the continuum, then the cardinality of any sphere and any horosphere in the $\mathbb{R}$-tree equals the continuum, and if the branch number of $\mathbb{R}$-tree is larger than the continuum, then the cardinality of any sphere or horosphere equals the number of branches.

Bibliographie

[1] Berestovskii V. N., Nikonorov Yu. G., Riemannian Manifolds and Homogeneous Geodesics, v. 4, SMI VSC RAS, Vladikavkaz, 2012, 414 pp. (in Russian) | MR

[2] Tits J., “A “ theorem of Lie-Kolchin” for trees”, Contributions to Algebra, Academic Press, N. Y., 1977, 377–388 | DOI | MR

[3] Chiswell I., Introduction to $\Lambda$-trees, Queen Mary Westfield College, University of London, U.K., 2001, 328 pp. | MR

[4] Alexandrov A. D., On a Generalization of Riemannian Geometry, Jahresber. Humb. Univ., Berlin, 1955

[5] Bridson M., Haefliger A., Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999, 643 pp. | DOI | MR | Zbl

[6] Dyubina A., Polterovich I., “Explicit constructions of universal $\mathbb{R}$-trees and asymptotic geometry of hyperbolic spaces”, Bull. Lond. Math. Soc., 33:6 (2001), 727–734 | DOI | MR | Zbl

[7] Berestovskii V. N., Plaut C., “Covering $\mathbb{R}$-trees, $\mathbb{R}$-free groups and dendrites”, Advances in Mathematics, 224:5 (2010), 1765–1783 | DOI | MR | Zbl

[8] Andreev P. D., “Semilinear Metric Semilattices on $\mathbb R$-Trees”, Russian Mathematics (Izvestiya VUZ. Matematika), 51:6 (2007), 1–10 | DOI | MR | Zbl

[9] Andreev P. D., Bulygin A. I., “On the Vertical Similarly Homogeneous $\mathbb{R}$-Trees”, Lobachevskii J. Math., 40:2 (2019), 127–139 | DOI | MR | Zbl

[10] Berestovskii V. N., “Similarly Homogeneous Locally Complete Spaces with an Intrinsic Metric”, Russian Mathematics (Izvestiya VUZ. Matematika), 48:11 (2004), 1–19 (in Russian) | Zbl

[11] Bestvina M., “$\mathbb{R}$-trees in topology, geometry and group theory”, Handbook of geometric topology, eds. R. J. Daverman, R. B. Sher, Elsevier Science, Amsterdam, 2002, 55–91 | MR | Zbl

[12] Aleksandrov A. D., “Mappings of Families of Sets”, Doklady Akademii Nauk SSSR, 190:3 (1970), 502–505 (in Russian) | Zbl

[13] Bogatyj S. A., Frolkina O. D., “Isometricity of Mappings Preserving Perimeter”, Vestnik Moskovskogo Universiteta. Ser. 1. Matematika Mekhanika, 2004, no. 1, 3–11 (in Russian)

[14] Berestovskii V. N., “Pathologies in Aleksandrov spaces of curvature bounded above”, Siber. Adv. Math., 12:4 (2002), 1–18 | MR | Zbl

[15] Kozyrev S. V., “Methods and Applications of Ultrametric and $p$-Adic Analysis: From Wavelet Theory to Biophysics”, Proceedings of the Steklov Institute of Mathematics, 274, suppl. 1 (2011), S1–S84 | DOI | DOI | MR | Zbl