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)

 
 

 

Geometric groups. I


Authors: Valera Berestovskii, Conrad Plaut and Cornelius Stallman
Journal: Trans. Amer. Math. Soc. 351 (1999), 1403-1422
MSC (1991): Primary 22D05, 53C21, 53C23, 53C70
DOI: https://doi.org/10.1090/S0002-9947-99-02086-3
Erratum: Tran. Amer. Math. Soc. 352 (2000), 5877
MathSciNet review: 1458295
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.


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

  • 1. Anderson, R.D., and Bing, R.H., A complete elementary proof that Hilbert space is homeomorphic to the countably infinite product of lines, Bull of the AMS 74 (1968) 771-792. MR 37:5847
  • 2. Berestovskii, V.N., Spaces with bounded curvature and distance geometry, Siberian Math. J. 27 (1986) 8-19. MR 88e:53110 (Russian original)
  • 3. Berestovskii, V. N., Homogeneous spaces with intrinsic metric, Soviet Math. Dokl. 27 (1989) 60-63.
  • 4. Berestovskii, V. N., The structure of locally compact spaces with an intrinsic metric, Siberian Math. J. 30 (1989) 23-34. MR 90c:53120
  • 5. Berestovskii, V. N., On Alexandrov's spaces with curvature bounded from above, Dokl. Rus. Akad. Nauk 324 (1995) 304-306.
  • 6. Berestovskii, V. N. and Plaut, C., Homogeneous spaces of curvature bounded below, to appear, J. Geometric Analysis.
  • 7. Bourbaki, N., Elements of Mathematics, General topology, Hermann, Paris, 1966. MR 34:5044b
  • 8. Bousfield, A. and Kan, D., Homotopy limits, completions and localizations, Lecture Notes in Math., no. 304, Springer, Berlin, 1972. MR 51:1825
  • 9. Brown, R., Higgins, P., and Morris, S., Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties, Math. Proc. Camb. Phil. Soc. 78 (1975) 19-32. MR 56:12168
  • 10. Cassels, J., and Fröhlich, A., Algebraic number theory, Thompson Book Company, Washington, D.C., 1967. MR 35:6500
  • 11. Chevalley, C., Theory of Lie Groups, Princeton Univeristy Press, Princeton, N.J., 1946. MR 18:583c
  • 12. Glushkov, V. M., Lie algebras of locally (bi)compact, Usp. Mat. Nauk 12 (1957) 137-142. MR 21:699
  • 13. Glushkov, V.M., On the structure of connected locally (bi)compact groups, Mat. Sbornik (N.S.) 48(90) (1959) 75-91. MR 23:A957
  • 14. Glushkov, V.M., The structure of locally compact groups and Hilbert's fifth problem, AMS Translations 15 (1960) 55-93. MR 22:5690
  • 15. Gromov, M., Lafontaine, F., and Pansu, P., Structures métriques pour les variétés riemannienes, Cedic, Paris, 1981. MR 85e:53051
  • 16. Helgason, S. Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. MR 80k:53081
  • 17. Husemoller, D., Fibre bundles, McGraw-Hill, New York, 1966. MR 37:4821
  • 18. Jacobson, N., Basic algebra II, Freeman, San Francisco, 1980. MR 81g:00001
  • 19. Iwasawa, K. On some types of topological groups, Ann. of Math. 50 (1949) 507-558. MR 10:679a
  • 20. Lashof, R., Lie algebras of locally compact groups, Pacific J. Math 7 (1957) 1145-1162. MR 19:1064a
  • 21. Montgomery, D. and Zippin, L., Topological transformation groups, Interscience, New York, 1955. MR 17:383b
  • 22. Munkres, J., Topology: a first course, Prentice-Hall, New Jersey, 1975. MR 57:4063
  • 23. Plaut, C., Geometrizing infinite-dimensional locally compact groups, Trans. of the AMS, 348 (1996) 941-962. MR 96h:53088
  • 24. Pontryagin, L. S., Topological groups, Gordon and Breach, New York, 1966. MR 34:1439
  • 25. Rickert, N., Arcs in locally compact groups, Math. Annalen 172 (1967) 222-228. MR 35:4331
  • 26. Spanier, E., Algebraic Topology, Springer, Berlin, 1966. MR 35:1007
  • 27. Stallman, C., Dissertation, University of Tennessee, 1996.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 22D05, 53C21, 53C23, 53C70

Retrieve articles in all journals with MSC (1991): 22D05, 53C21, 53C23, 53C70


Additional Information

Valera Berestovskii
Affiliation: Department of Mathematics, Omsk State University, Pr. Mira 55A, Omsk 77 644077 Russia
Email: berest@univer.omsk.su

Conrad Plaut
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: plaut@novell.math.utk.edu

Cornelius Stallman
Affiliation: Department of Mathematics and Computer Science, Augusta State University, Augusta, Georgia 30904-2200

DOI: https://doi.org/10.1090/S0002-9947-99-02086-3
Keywords: Topological groups, Lie groups, invariant metrics, Alexandrov curvature bounded below, universal covering group
Received by editor(s): February 14, 1997
Additional Notes: The paper was partly written while the first author was visiting the University of Tennessee, and he wishes to acknowledge the support of the Tennessee Science Alliance and the Mathematics Department.
The second and third authors were partly supported by NSF grant DMS-9401302, and the second by a UTK Faculty Development Award
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society