Characterizing Lie groups by controlling their zero-dimensional subgroups
-
Dikran Dikranjan
Abstract
We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows:
(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.
(ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete.
(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.
(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 22540089
Award Identifier / Grant number: L13710
Funding statement: The first named author gratefully acknowledges a JSPS long term visitor grant L13710, as well as the generous hospitality of the Division of Mathematics, Physics and Earth Sciences at the Graduate School of Science and Engineering of Ehime University, Matsuyama. The second named author was partially supported by the Grant-in-Aid for Scientific Research (C) No. 22540089 by the Japan Society for the Promotion of Science (JSPS).
References
[1] F. Ancel, T. Dobrowolski and J. Grabowski, Closed subgroups in Banach spaces, Studia Math. 109 (1994), 277–290. Suche in Google Scholar
[2] A. V. Arhangel’skii, Cardinal invariants of topological groups. Embeddings and condensations (in Russian), Dokl. Akad. Nauk SSSR 247 (1979), 779–782; translation in Soviet Math. Dokl. 20 (1979), 783–787. Suche in Google Scholar
[3] A. V. Arhangel’skii, Spaces of functions in the topology of pointwise convergence, and compacta (in Russian), Uspekhi Mat. Nauk 39 (1984), no. 5(239), 11–50; translation in Russian Math. Surveys 39 (1984), no. 5, 9–56. Suche in Google Scholar
[4] L. Außenhofer, M. J. Chasco, D. Dikranjan and X. Dominguez, Locally minimal topological groups 1, J. Math. Anal. Appl. 370 (2010), 431–452. 10.1016/j.jmaa.2010.04.026Suche in Google Scholar
[5] L. Außenhofer, M. J. Chasco, D. Dikranjan and X. Dominguez, Locally minimal topological groups 2, J. Math. Anal. Appl. 380 (2011), no. 2, 552–570. 10.1016/j.jmaa.2011.03.056Suche in Google Scholar
[6] H. Bohr, Collected Mathematical Works, Dansk Matemetitisk Forsening, Copenhagen, 1952. Suche in Google Scholar
[7] J. Cleary and S. Morris, Topologies on locally compact groups, Bull. Aust. Math. Soc. 38 (1988), 105–111. 10.1017/S0004972700027313Suche in Google Scholar
[8] W. W. Comfort, K. H. Hofmann and D. Remus, A survey on topological groups and semigroups, Recent Progress in General Topology, North Holland, Amsterdam (1992), 58–144. Suche in Google Scholar
[9] H. F. Davis, A note on Haar measure, Proc. Amer. Math. Soc. 6 (1955), 318–321. 10.1090/S0002-9939-1955-0069187-XSuche in Google Scholar
[10] D. Dikranjan, On a class of finite-dimensional compact abelian groups, Topology Theory and Applications (Eger 1983), Colloq. Math. Soc. János Bolyai 43, János Bolyai Mathematical Society, Budapest (1983), 215–231. Suche in Google Scholar
[11] D. Dikranjan, Density and total density of the torsion part of a compact group, Rend. Accad. Naz. Sci. Detta XL V. Ser. Mem. Mat. 14 (1990), no. 1, 235–252. Suche in Google Scholar
[12] D. Dikranjan, Zero-dimensionality of some pseudocompact groups, Proc. Amer. Math. Soc. 120 (1994), 1299–1308. 10.1090/S0002-9939-1994-1185278-9Suche in Google Scholar
[13] D. Dikranjan, Countably compact groups satisfying the open mapping theorem, Topology Appl. 98 (1999), 81–129. 10.1016/S0166-8641(99)00047-4Suche in Google Scholar
[14] D. Dikranjan, W. He, Z. Xiao and W. Xi, Locally compact groups and locally minimal group topologies, to appear in Fund. Math. Suche in Google Scholar
[15] D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology III, Springer, Berlin (2014), 229–327. 10.2991/978-94-6239-024-9_6Suche in Google Scholar
[16] D. Dikranjan and I. Prodanov, Totally minimal topological groups, God. Sofij. Univ. Fak. Mat. Mekh. 69(1974/75) (1979), 5–11. Suche in Google Scholar
[17] D. Dikranjan and I. Prodanov, A class of compact abelian groups, God. Sofij. Univ. Fak. Mat. Mekh. 70(1975/76) (1981), 191–206. Suche in Google Scholar
[18] D. Dikranjan, I. Prodanov and L. Stoyanov, Topological Groups: Characters, Dualities and Minimal Group Topologies, Pure Appl. Math. 130, Marcel Dekker, New York, 1989. Suche in Google Scholar
[19] D. Dikranjan and D. Shakhmatov, Compact-like totally dense subgroups of compact groups, Proc. Amer. Math. Soc. 114 (1992), 1119–1129. 10.1090/S0002-9939-1992-1081694-2Suche in Google Scholar
[20] D. Dikranjan and D. Shakhmatov, Algebraic structure of pseudocompact groups, Mem. Amer. Math. Soc. 633 (1998), 1–83. 10.1090/memo/0633Suche in Google Scholar
[21] D. Dikranjan and D. Shakhmatov, Selected topics from the structure theory of topological groups, Open Problems in Topology II, Elsevier, Amsterdam (2007), 389–406. 10.1016/B978-044452208-5/50041-7Suche in Google Scholar
[22] D. Dikranjan, D. Shakhmatov and J. Spěvák, NSS and TAP properties in topological groups closed to being compact, preprint (2009), https://arxiv.org/abs/0909.2381. Suche in Google Scholar
[23] D. Dikranjan, D. Shakhmatov and J. Spěvák, Productivity of sequences with respect to a given weight function, Topology Appl. 158 (2011), 298–324. 10.1016/j.topol.2010.11.009Suche in Google Scholar
[24] D. Dikranjan and L. Stoyanov, Compact groups with totally dense torsion part, Baku International Topological Conference (Baku 1987), Ehlm, Baku (1989), 231–238. Suche in Google Scholar
[25] D. Dikranjan and M. Tkachenko, Sequential completeness of quotient groups, Bull. Aust. Math. Soc. 61 (2000), 129–151. 10.1017/S0004972700022085Suche in Google Scholar
[26] D. Dikranjan and M. Tkachenko, Sequentially complete groups: Dimension and minimality, J. Pure Appl. Algebra 157 (2001), 215–239. 10.1016/S0022-4049(00)00015-3Suche in Google Scholar
[27] T. Dobrowolski and J. Grabowski, Subgroups of Hilbert spaces, Math. Z. 211 (1992), 657–669. 10.1007/BF02571453Suche in Google Scholar
[28] D. Doïtchinov, Produits de groupes topologiques minimaux, Bull. Sci. Math. 97 (1972), 59–64. Suche in Google Scholar
[29] P. Flor, Zur Bohr-Konvergenz von Folgen, Math. Scand. 23 (1968), 169–170. 10.7146/math.scand.a-10907Suche in Google Scholar
[30] L. Fuchs, Infinite Abelian Groups. Vol. I, Academic Press, New York, 1970. Suche in Google Scholar
[31] L. Fuchs, Infinite Abelian Groups. Vol. II, Academic Press, New York, 1973. Suche in Google Scholar
[32] J. Galindo and S. Macario, Pseudocompact group topologies with no infinite compact subsets, J. Pure Appl. Algebra 215 (2011), 655–663. 10.1016/j.jpaa.2010.06.014Suche in Google Scholar
[33] I. Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269–276. 10.4153/CJM-1962-017-3Suche in Google Scholar
[34] V. M. Glushkov, Structure of locally bicompact groups and Hilbert’s fifth problem (in Russian), Uspekhi Mat. Nauk 12 (1957), 3–41. Suche in Google Scholar
[35] I. Guran, Topological groups similar to Lindelöf groups (in Russian), Dokl. Akad. Nauk SSSR 256 (1981), 1305–1307; translation in Soviet Math. Dokl. 23 (1981), 173–175. Suche in Google Scholar
[36] S. Hérnandez, K.-H. Hofmann and S. Morris, The weights of closed subgroups of a locally compact group, J. Group Theory 15 (2012), 613–630. 10.1515/jgt-2012-0012Suche in Google Scholar
[37] K.-H. Hofmann and S. A. Morris, The Structure of Compact Groups. A Primer for the Student — A Handbook for the Expert. Second Revised and Augmented Edition, De Gruyter Stud. Math. 25, Walter de Gruyter, Berlin, 2006. 10.1515/9783110199772Suche in Google Scholar
[38] D. Lee, Supplements for the identity component in locally compact groups, Math. Z. 104 (1968), 28–49. 10.1007/BF01114916Suche in Google Scholar
[39] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955. Suche in Google Scholar
[40] S. Morris and V. Pestov, On Lie groups in varieties of topological groups, Colloq. Math. 78 (1998), no. 1, 39–47. 10.4064/cm-78-1-39-47Suche in Google Scholar
[41] I. Prodanov, Precompact minimal group topologies and p-adic numbers, God. Sofij. Univ. Mat. Fak. 66(1971/72) (1974), 249–266. Suche in Google Scholar
[42] D. Shakhmatov, Imbeddings into topological groups preserving dimensions, Topology Appl. 36 (1990), 181–204. 10.1016/0166-8641(90)90008-PSuche in Google Scholar
[43] D. Shakhmatov, On zero-dimensionality of subgroups of locally compact groups, Comment. Math. Univ. Carolin. 32 (1991), no. 3, 581–582. Suche in Google Scholar
[44] D. Shakhmatov and J. Spěvák, Group-valued continuous functions with the topology of pointwise convergence, Topology Appl. 157 (2010), 1518–1540. 10.1016/j.topol.2009.06.022Suche in Google Scholar
[45] S. M. Sirota, The product of topological groups and extremal disconnectedness, Math. USSR Sbornik 8 (1969), 169–180. 10.1070/SM1969v008n02ABEH001115Suche in Google Scholar
[46] R. M. Stephenson, Jr., Minimal topological groups, Math. Ann. 192 (1971), 193–195. 10.1007/BF02052870Suche in Google Scholar
[47] M. Tkachenko, Countably compact and pseudocompact topologies on free abelian groups (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1990 (1990), no. 5(336), 68–75; translation in Soviet Math. (Iz. VUZ) 34 (1990), no. 5, 79–86. Suche in Google Scholar
[48] M. Tkachenko, On dimension of locally pseudocompact groups and their quotients, Comment. Math. Univ. Carolin. 31 (1990), 159–166. Suche in Google Scholar
[49] M. Tkachenko, Topological groups in which all countable subgroups are closed, Topology Appl. 159 (2012), 1806–1814. 10.1016/j.topol.2011.11.054Suche in Google Scholar
[50] D. van Dantzig, Studien over topologische algebra, Dissertation, Amsterdam, 1931. Suche in Google Scholar
[51] E. van Douwen, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (1980), 417–427. 10.1090/S0002-9947-1980-0586725-8Suche in Google Scholar
[52] N. T. Varopoulos, A theorem on the continuity of homomorphisms of locally compact groups, Proc. Camb. Phil. Soc. 60 (1964) 449–463. 10.1017/S0305004100037968Suche in Google Scholar
[53] H. Wilcox, Dense subgroups of compact groups, Proc. Amer. Math. Soc. 28 (1971), 578–580. 10.1090/S0002-9939-1971-0280640-5Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- The variance of divisor sums in arithmetic progressions
- Characterizing Lie groups by controlling their zero-dimensional subgroups
- CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
- Towards a Goldberg–Shahidi pairing for classical groups
- On the non-existence of cyclic splitting fields for division algebras
- Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
- Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
- Anharmonic solutions to the Riccati equation and elliptic modular functions
- A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
- Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
- On symplectic semifield spreads of PG(5,q2), q odd
- From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
- Golodness and polyhedral products for two-dimensional simplicial complexes
Artikel in diesem Heft
- Frontmatter
- The variance of divisor sums in arithmetic progressions
- Characterizing Lie groups by controlling their zero-dimensional subgroups
- CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups
- Towards a Goldberg–Shahidi pairing for classical groups
- On the non-existence of cyclic splitting fields for division algebras
- Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
- Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients
- Anharmonic solutions to the Riccati equation and elliptic modular functions
- A non-homogeneous local Tb theorem for Littlewood–Paley g*λ-function with Lp-testing condition
- Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
- On symplectic semifield spreads of PG(5,q2), q odd
- From Freudenthal’s spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra
- Golodness and polyhedral products for two-dimensional simplicial complexes
