The (topological) isomorphism of the automorphism group of a homogeneous chain $X$ and the (topological) wreath product of automorphism groups of a regular interval $J$ with respect to the group of automorphisms of the quotient space $X∕J$ is established. A characterization of the Roelcke-precompactness of automorphism groups of general chains is given. The equivalence of the Roelcke precompactness of the automorphism group of a general chain in permutation topology and pointwise convergence topology in the presence of a simple proper regular interval is established.
Keywords: Roelcke precompactness, homogeneous chains, automorphisms group, wreath product
Received December 14, 2024
Revised January 23, 2025
Accepted January 27, 2025
Boris Vladimirovich Sorin, Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Dept. of General Topology and Geometry, Moscow, 119991 Russia, e-mail: bvs@imtprofi.ru
REFERENCES
1. Roelcke W., Dierolf S. Uniform structures on topological groups and their quotients. NY, McGraw-Hill Inc., 1981, 276 p. ISBN-10: 0070534128 .
2. Uspenskij V.V. Compactifications of topological groups. In: Proc. Ninth Prague Topological Symposium, Prague, Czech Republic, 2001, pp. 331–346. https://doi.org/10.48550/arXiv.math/0204144
3. Pestov V. Topological groups: Where to from here? Topology Proc., 1999, vol. 24, pp. 421–506. https://doi.org/10.48550/arXiv.math/9910144
4. Tsankov T. Unitary representations of oligomorphic groups. Geom. Funct. Anal., 2012, vol. 22, no. 2, pp. 528–555. https://doi.org/10.1007/s00039-012-0156-9
5. Sorin B.V. The Roelcke precompactness and compactifications of transformations groups of discrete spaces and homogeneous chains. 2024, 24 p. Available at: https://arxiv.org/abs/2310.18570v2 .
6. Gaughan E.D. Topological group structures of infinite symmetric groups. Proc. Nat. Acad. Sci. USA, 1976, vol. 58, no. 3, pp. 907–910. https://doi.org/10.1073/pnas.58.3.907
7. Ovchinnikov S. Topological automorphism group of chains. Mathwear and Soft Comp., 2001, vol. 8, pp. 47–60.
8. Sorin B.V. Compactifications of homeomorphism groups of linearly ordered compacta. Math Notes, 2022, vol. 112, no. 1, pp. 126–141. https://doi.org/10.1134/S0001434622070148
9. Glasner E., Megrelishvili M. Circular orders, ultra-homogeneous order structures and their automorphism groups in topology. Chapter in: Topology, geometry, and dynamics: V. A. Rokhlin-Memorial, Contemp. Math., eds.: A. M. Vershik, V. M. Buchstaber, A. V. Malyutin, 2021, vol. 772, pp. 133–154. https://doi.org/10.48550/arXiv.1803.06583
10. Ohkuma T. Structure of homogeneous chains. Kodai Math. Sem. Rep., 1953, vol. 5, no. 1, pp. 1–12. https://doi.org/10.2996/kmj/1138843293
11. Engelking R. General topology, 2nd edt. Sigma Ser. Pure Math., vol. 6, Berlin, Heldermann Verlag, 1989, 529 p. ISBN: 9783885380061 .
12. Kuratowski K., Mostowski A. Set theory. Warszawa, PWN Polish Scient. Publ., 1967. Translated to Russian under the title Teoriya mnozhestv, Moscow, Mir Publ., 1970, 416 p.
13. Ohkuma T. On discrete homogeneous chains. Kodai Math. Sem. Rep., 1952, vol. 4, no. 1, pp. 23–30. https://doi.org/10.2996/kmj/1138843210
14. Glass A.M.W. Groups of automorphisms of totally ordered sets: techniques, model theory and applications to decision problems. In: Groups, modules, and model theory — Surveys and recent developments. Droste M., Fuchs L., Goldsmith B., StrЈungmann L. (eds). Cham, Springer, 2017, pp. 109–134. https://doi.org/10.1007/978-3-319-51718-6_6
15. McCleary S.H. Lattice-ordered permutation groups: the structure theory. In: Ordered groups and infinite permutation groups. Mathematics and its applications, vol. 354. Holland W.C. (eds). Springer, Boston, MA, 1996, pp. 29–62. https://doi.org/10.1007/978-1-4613-3443-9_2
16. Glass A.M.W., Gurevich Yu., Holland W.C., Shelah S. Rigid homogeneous chains. Math. Proc. Camb. Phil. Soc., 1981, vol. 89, no. 7, pp. 7–17. https://doi.org/10.1017/S0305004100057881
17. Holland W.C. Transitive lattice-ordered permutations groups. Math. Z., 1965, vol. 87, pp. 420–433. https://doi.org/10.1007/BF01111722
18. Arens R. Topologies for homeomorphism groups. Amer. J. Math., 1946, vol. 68, no. 4, pp. 593–610. https://doi.org/10.2307/2371787
19. Kozlov K.L. Uniform equicontinuity and groups of homeomorphisms. Topol. Appl., 2022, vol. 311, art. no. 107959. https://doi.org/10.1016/j.topol.2021.107959
20. Holland W.C., McCleary S.H. Wreath products of ordered permutations groups. Pacific J. Math., 1969, vol. 31, no. 3, pp. 703–716.
Cite this article as: B.V. Sorin. On Roelcke precompactness of automorphism groups of general chains. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 175–184.
