ИСТИНА

The Gromov–Hausdorff (GH-) distance between two metric spaces X and Y is a measure of difference between these spaces. To be more precise, let us isometrically embed X and Y into other metric spaces Z, in all possible ways, and calculate the least possible Hausdorff distance between the images. The resulting value is called the GH-distance between X and Y [1]. There are various beautiful applications of this notion like Gromov’s theorem on groups of polynomial growth [2] and Gromov’s compactness theorem [3, 4]. The most common use of the distance is related to description of the corresponding convergence. In the present talk we shall speak about another aspect. Namely, we discuss the geometry and topology of the first natural space endowed with the GH-distance, namely, the space M of isometry classes of compact metric spaces. It is well-know that GH-distance on M is a metric, and M with this metric is usually called the Gromov–Hausdorff space. We shall discuss the both classical and recent results devoted to M, in particular, its local and global symmetries. For more details see [5]–[15]. References [1] D.Yu. Burago, Yu.D. Burago, S.V. Ivanov. A Course in Metric Geometry // AMS, Providence, RI, 2001. [2] M. Gromov, Groups of polynomial growth and expanding maps // Inst. Hautes Etudes Sci. Publ. Math., 53 (1981), 53–73. [3] M. Gromov, Structures m´etriques pour les vari´et´es riemanniennes // CEDIC/Fernand Nathan (1981). [4] J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. 46:3 (1997), 406–480. [5] A.O. Ivanov, N.K. Nikolaeva, A.A. Tuzhilin. The Gromov–Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic // ArXiv e-prints, 1504.03830 (2015). [6] A.O. Ivanov, S. Iliadis, A.A. Tuzhilin, Realizations of Gromov-Hausdorff Distance // ArXiv e-prints, 1603.08850 (2016). [7] A.O. Ivanov, N.K. Nikolaeva, A.A. Tuzhilin, Steiner Problem in Gromov-Hausdorff Space: the Case of Finite Metric Spaces // ArXiv e-prints, 1604.02170 (2016). [8] A.O. Ivanov, A.A. Tuzhilin, Gromov–Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings // ArXiv e-prints, 1604.06116 (2016). [9] A.O. Ivanov, S. Iliadis, A.A. Tuzhilin, Local Structure of Gromov-Hausdorff Space, and Isometric Embeddings of Finite Metric Spaces into this Space // ArXiv e-prints, 1604.07615 (2016). [10] A.O. Ivanov, A.A. Tuzhilin, Steiner Ratio and Steiner-Gromov Ratio of Gromov-Hausdorff Space // ArXiv e-prints, 1605.01094 (2016). [11] A.A. Tuzhilin, Calculation of Minimum Spanning Tree Edges Lengths using Gromov–Hausdorff Distance // ArXiv e-prints, 1605.01566 (2016). [12] A.O. Ivanov, A.A. Tuzhilin, Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes // ArXiv e-prints, 1607.06655 (2016). [13] A.O. Ivanov, A.A. Tuzhilin, Local Structure of Gromov–Hausdorff Space near Finite Metric Spaces in General Position // ArXiv e-prints, 1611.04484 (2016). [14] A.A. Tuzhilin, Who Invented the Gromov-Hausdorff Distance? // ArXiv e-prints, 1612.00728 (2016). [15] A.O. Ivanov, A.A. Tuzhilin, Isometry Group of Gromov–Hausdorff Space // ArXiv e-prints, 1806.02100 (2018).