Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On centralizers of generalized uniform subgroups of locally compact groups


Author: Kwan-Yuk Law Sit
Journal: Trans. Amer. Math. Soc. 201 (1975), 133-146
MSC: Primary 22D05
DOI: https://doi.org/10.1090/S0002-9947-1975-0354923-2
MathSciNet review: 0354923
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a locally compact group and $ H$ a closed subgroup of $ G$ such that the homogeneous space $ G/H$ admits a finite invariant measure. Let $ {Z_G}(H)$ be the centralizer of $ H$ in $ G$. It is shown that if $ G$ is connected then $ {Z_G}(H)$ modulo its center is compact. If $ G$ is only assumed to be locally connected it is shown that the commutator subgroup of $ {Z_G}(H)$ has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group $ G$ is given to show that the results for $ {Z_G}(H)$ need not hold if $ G$ is not connected or locally connected.


References [Enhancements On Off] (What's this?)

  • [1] A Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179-188. MR 23 #A964. MR 0123639 (23:A964)
  • [2] H. Garland and M. Goto, Lattices and the adjoint group of a Lie group, Trans. Amer. Math. Soc. 124 (1966), 450-460. MR 33 #7459. MR 0199311 (33:7459)
  • [3] F. Greenleaf, M. Moskowitz and L. P. Rothschild, Compactness of certain homogeneous spaces of finite volume, Amer. J. Math. (to appear). MR 0407196 (53:10979)
  • [4] S. Grosser, O. Loos and M. Moskowitz, Über Automorphismengruppen lokal-kompakter Gruppen und Derivationen von Lie-Gruppen, Math. Z. 114 (1970), 321-339. MR 41 #8575. MR 0263976 (41:8575)
  • [5] S. Grosser and M. Moskowitz, On central topological groups, Trans. Amer. Math. Soc. 127 (1967), 317-340. MR 35 #292. MR 0209394 (35:292)
  • [6] -, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1-40. MR 44 #1766. MR 0284541 (44:1766)
  • [7] E. Hewitt and K. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
  • [8] G. Hochschild, The structure of Lie groups, Holden-Day, San Francisco, 1965. MR 34 #7696. MR 0207883 (34:7696)
  • [9] K. Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507-558. MR 10, 679. MR 0029911 (10:679a)
  • [10] A. G. Kuroš, Theory of groups, 2nd ed., GITTL, Moscow, 1953; English transl., Chelsea, New York, 1955. MR 15, 501; 17, 124.
  • [11] D. H. Lee, On the centralizer of a subgroup of a Lie group, Proc. Amer. Math. Soc. 30 (1971), 195-198. MR 44 #367. MR 0283134 (44:367)
  • [12] J. R. Liukkonen, Dual spaces of groups with precompact conjugacy classes, Trans. Amer. Math. Soc. 180 (1973), 85-108. MR 0318390 (47:6937)
  • [13] G. D. Mostow, Homogeneous spaces with finite invariant measure, Ann. of Math. (2) 75 (1962), 17-37. MR 26 #2546. MR 0145007 (26:2546)
  • [14] B. H. Neumann, Groups with finite classes of conjugacy elements, Proc. London Math. Soc. (3) 1 (1951), 178-187. MR 13, 316. MR 0043779 (13:316c)
  • [15] L. C. Robertson, A note on the structure of Moore groups, Bull. Amer. Math. Soc. 75 (1969), 594-599. MR 39 #7027. MR 0245721 (39:7027)
  • [16] K. Y. L. Sit, On bounded elements and centralizers of generalized uniform subgroups of locally compact groups, Doctoral thesis, The City Univ. of New York, New York, 1973.
  • [17] R. Tolimieri, On the Selberg conditions for subgroups of solvable Lie groups, Bull. Amer. Math. Soc. 77 (1971)) 584-586. MR 45 #448. MR 0291355 (45:448)
  • [18] H. C. Wang, On the deformation of a lattice in a Lie group, Amer. J. Math. 85 (1963), 189-212. MR 27 #2582. MR 0152606 (27:2582)
  • [19] S. P. Wang, On the centralizer of a lattice, Proc. Amer. Math. Soc. 21 (1969), 21-23. MR 38 #5989. MR 0237708 (38:5989)
  • [20] -, On $ S$-subgroups of solvable Lie groups, Amer. J. Math. 92 (1970), 389-397. MR 41 #8581. MR 0263982 (41:8581)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22D05

Retrieve articles in all journals with MSC: 22D05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0354923-2
Keywords: Locally compact group, Lie group, periodic subset, compactness conditions, homogeneous space, invariant measure, Borel's density theorem, lattice
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society