Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


We survey recent results on connections of mappings with bounded distortion and dynamics of discrete action in different geometries. Our new tool based on quasiconformal and conformal dynamics of discrete group actions in 3-geometries at infinity of negatively curved symmetric rank one spaces is used to construct new types of quasiconformal, quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings, as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices.

Leaving applications of our approach to geometry and topology of manifolds to another our papers, here we discuss applications of our constructions to long standing problems for mappings with bounded distortion in different geometries, especially M.A.Lavrentiev surjectivity problem in Euclidean space, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball, as well as a possibility of such applications in spaces with complex and spherical CR-structures.

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1. E.M. Andreev, On convex polyhedra in Lobachevsky space, Mat. Sbornik 81, 1970, 445–478 (In Russian); Engl. Transl.: Math. USSR Sbornik 10, 1970, 413–440.

2. Boris Apanasov, Nontriviality of Teichmüller space for Kleinian group in space. - In: Riemann Surfaces and Related Topics, Proc. 1978 Stony Brook Conference (I.Kra and B.Maskit, eds), Ann. of Math. Studies 97, Princeton Univ. Press, 1981, 21–31.

3. Boris Apanasov, Conformal geometry of discrete groups and manifolds. - De Gruyter Exp. Math. 32, W. de Gruyter, Berlin - New York, 2000.

4. Boris Apanasov, Quasisymmetric embeddings of a closed ball inextensible in neighborhoods of any boundary points. - Ann. Acad. Sci. Fenn., Ser. A I Math 14, 1989, 243–255.

5. Boris Apanasov, Nonstandard uniformized conformal structures on hyperbolic manifolds. - Invent. Math. 105, 1991, 137–152.

6. Boris Apanasov, Hyperbolic 4-cobordisms and group homomorphisms with infinite kernel. - Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 57, 2010, 31–44.

7. Boris Apanasov, Group Actions, Teichmüller Spaces and Cobordisms. - Lobachevskii J. Math., 38, 2017, 213-228.

8. Boris Apanasov, Topological barriers for locally homeomorphic quasiregular mappings in 3-space. - Ann. Acad. Sci. Fenn. Math. 43, 2018, 579–596.

9. Boris Apanasov, Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space. - (Bogdan Bojarski Memorial Volume), J. Math. Sci. 242, 2019, 760–771 (Ukrain. Math. Bull. 16, 2019, 10–27).

10. Boris Apanasov, Non-faithful discrete representations of hyperbolic lattices, hyperbolic 4-cobordisms and applications. - Prepr. Univ. of Oklahoma, 2021, 25 pp.

11. Boris Apanasov, Conformal interbreeding, Teichmüller spaces and applications. - In: Topology and Geometry: Extremal and Typical, Univ. of Chicago and Univ. of California at Santa Barbara, 2021, 1-28.

12. Boris Apanasov, Quasiregular mappings and discrete group actions.- J. Math. Sci. 260, 2022, 601-618.

13. Boris Apanasov, Dynamics of discrete group action. - De Gruyter Advances in Anal. and Geom., in progress.

14. Boris Apanasov, Bending deformations of complex hyperbolic surfaces. - J. für die reine und angewandte Math., 492 (1997), 75-91.

15. Boris Apanasov, Geometry and topology of complex hyperbolic and CR-manifolds. - Uspekhi Mat. Nauk 52, No. 5 (1997), 9-41 (Russian); Engl. Transl.: Russian Math. Surveys, 52 (1998), 895-928.

16. Boris Apanasov, Quasiconformally instable disc bundles with complex structures. - Proceedings of the Steklov Institute of Mathematics, 252 (2006), 1-11.

17. Boris Apanasov and Inkang Kim, Cartan’s angular invariant and deformations in symmetric spaces of rank 1. - Mat. Sbornik 198 (2007), no. 2, 3–28 (Russian); Engl.Transl.: Sbornik Math. 198 (2007), no. 1-2, 147–169

18. Boris Apanasov and Andrei V. Tetenov, Nontrivial cobordisms with geometrically finite hyperbolic structures. - J. of Diff. Geom. 28, 1988, 407–422.

19. Boris Apanasov and Xiangdong Xie, Geometrically finite complex hyperbolic manifolds. - Intern. J. of Math., 8 (1997), 703-757.

20. Boris Apanasov and Xiangdong Xie, Discrete actions on nilpotent groups and negatively curved spaces. - J. Diff. Geometry Appl., 20(2004), 11-29.

21. Vladislav V. Aseev, Quasi-symmetric embeddings. - Complex analysis and representation theory, 3. - J. Math. Sci. 108 (2002), 375–410.

22. V.V. Aseev, A.V. Sychev and A.V. Tetenov, Gluing of quasisymmetric embeddings in the problem of quasi-conformal extension. - Ukr. Mat. Zh. 56 (2004), no. 6, 737–744; (Russian); Engl.Transl.: Ukrainian Math. J. 56 (2004), no. 6, 873–882.

23. V.V. Aseev, D.G. Kuzin and A.V. Tetenov, Angles between sets and the gluing of quasisymmetric mappings in metric spaces. - Izv. Vyssh. Uchebn. Zaved. Mat. 10 (2005), 3–13; (Russian); Engl.Transl.: Russian Math. (Iz. VUZ) 49 (2006), no. 10, 1–10.

24. K.F. Barth, D.A. Brannan and W.K. Hayman, Research problems in complex analysis, Bull. London Math. Soc. 16, 1984, no. 5, 490–517.

25. Dan Burns and Rafe Mazzeo, On the geometry of cusps for SU(n,1). - Manifolds and Geometry (Pisa, 1993), Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996, 112–131.

26. James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123–148.

27. Kevin Corlette, Hausdorff dimensions of limit sets. I. - Invent. Math. 102 (1990), 521–541.

28. Kevin Corlette and Alessandra Iozzi, Limit sets of discrete groups of isometries of exotic hyperbolic spaces. - Trans. Amer. Math. Soc. 351 (1999), 1507–1530.

29. C. Epstein, R. Melrose and G. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains. - Acta Math. 167(1991), 1-106.

30. Pierre Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335–400.

31. William M. Goldman and John R. Parker, Complex hyperbolic ideal triangle groups. - J. Reine Angew. Math. 425 (1992), 71–86.

32. Rostislav Grigorchuk, Milnor’s problem on the growth of groups and its consequences. - In: Frontiers in complex dynamics, Princeton Math. Ser. 51, Princeton Univ. Press, 2014, 705–773.

33. Mikhael Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, Boston, 1999.

34. Mikhael Gromov and Ilia I. Piatetski-Shapiro, Non-arithmetic groups in Lobachevsky spaces, Publ. Math. IHES 66, 1988, 93-103.

35. W.K.Hayman and E.F. Lingham, Research problems in function theory. - ArXiv: 1809.07200

36. J.J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold. - Ann. of Math., 81(1965), 451-472.

37. Mikhael A. Lavrentiev, On a class of continuous mappings, Mat. Sbornik, 42, 1935, 407–424. (Russian).

38. Olli Martio, Seppo Rickman and Jussi Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math 465, 1970, 1–13.

39. Oleg R. Musin, The kissing number in four dimensions. - Ann. of Math. 168 (2008), 1–32.

40. Terrence Napier and Mohan Ramachandran, Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems. - Geom. Funct. Anal. 5 (1995), no. 5, 809–851.

41. Terrence Napier and Mohan Ramachandran, The L 2 ∂-method, weak Lefschetz theorems, and topology of Kähler manifolds. - J. Amer. Math. Soc. 11 (1998), 375–396.

42. Kai Rajala, Radial limits of quasiregular local homeomorphisms, Amer. J. Math. 130, 2008, 269–289.

43. Yuri G. Reshetnyak, Space mappings with bounded distortion. - Nauka, Novosibirsk, 1982 (Russian); Engl. Transl.: Transl. Math. Monogr. 73, Amer. Math. Soc., Providence, 1989.

44. Seppo Rickman, Quasiregular Mappings. - Ergeb. Math. Grenzgeb. 26, Springer, Berlin–Heidelberg, 1993.

45. Richard E. Schwartz, A better proof of the Goldman-Parker conjecture. - Geom. Topol. 9 (2005), 1539–1601.

46. Dennis Sullivan, Hyperbolic geometry and homeomorphisms. - In: Geometric Topology (J.C.Cantrell, Ed.), Academic Press, New York, 1979, 543-555.

47. Dennis Sullivan, Quasiconformal homeomorphisms and dynamics, II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), 243-260.

48. William Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. AMS 6, 1982, 357–381.

49. Pekka Tukia, On isomorphisms of geometrically finite Kleinian groups, Publ. Math. IHES 61, 1985, 171–214.

50. Matti Vuorinen, Cluster sets and boundary behavior of quasiregular mappings. Math. Scand. 45, 1979, no. 2, 267–281.

51. Matti Vuorinen, Queries No 249. - Notices Amer.Math.Soc. 28, 1981, no. 7, 607.

52. Matti Vuorinen, Conformal geometry and quasiregular mappings. - Lecture Notes in Math 1319, Springer, Berlin-Heidelberg, 1988.



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