## Geometric groups. I

HTML articles powered by AMS MathViewer

- by Valera Berestovskii, Conrad Plaut and Cornelius Stallman PDF
- Trans. Amer. Math. Soc.
**351**(1999), 1403-1422 Request permission

Erratum: Trans. Amer. Math. Soc.

**352**(2000), 5877.

## 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

- R. D. Anderson and R. H. Bing,
*A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines*, Bull. Amer. Math. Soc.**74**(1968), 771–792. MR**230284**, DOI 10.1090/S0002-9904-1968-12044-0 - V. N. Berestovskiĭ,
*Spaces with bounded curvature and distance geometry*, Sibirsk. Mat. Zh.**27**(1986), no. 1, 11–25, 197 (Russian). MR**847410** - Berestovskii, V. N.,
*Homogeneous spaces with intrinsic metric*, Soviet Math. Dokl.**27**(1989) 60-63. - V. N. Berestovskiĭ,
*The structure of locally compact homogeneous spaces with an intrinsic metric*, Sibirsk. Mat. Zh.**30**(1989), no. 1, 23–34 (Russian); English transl., Siberian Math. J.**30**(1989), no. 1, 16–25. MR**995016**, DOI 10.1007/BF01054211 - Berestovskii, V. N.,
*On Alexandrov’s spaces with curvature bounded from above,*Dokl. Rus. Akad. Nauk 324 (1995) 304-306. - Berestovskii, V. N. and Plaut, C.,
*Homogeneous spaces of curvature bounded below,*to appear, J. Geometric Analysis. - Nicolas Bourbaki,
*Elements of mathematics. General topology. Part 1*, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR**0205210** - A. K. Bousfield and D. M. Kan,
*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573**, DOI 10.1007/978-3-540-38117-4 - Ronald Brown, Philip J. Higgins, and Sidney A. Morris,
*Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties*, Math. Proc. Cambridge Philos. Soc.**78**(1975), 19–32. MR**453915**, DOI 10.1017/S0305004100051483 - J. W. S. Cassels and A. Fröhlich (eds.),
*Algebraic number theory*, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR**0215665** - Cahit Arf,
*Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper*, J. Reine Angew. Math.**181**(1939), 1–44 (German). MR**18**, DOI 10.1515/crll.1940.181.1 - V. M. Gluškov,
*Lie algebras of locally bicompact groups*, Uspehi Mat. Nauk (N.S.)**12**(1957), no. 2 (74), 137–142 (Russian). MR**0101893** - V. M. Gluškov,
*On the structure of connected locally bicompact groups*, Mat. Sb. (N.S.)**48 (90)**(1959), 75–91 (Russian). MR**0123632** - V. M. Gluškov,
*The structure of locally compact groups and Hilbert’s fifth problem.*, Amer. Math. Soc. Transl. (2)**15**(1960), 55–93. MR**0114872**, DOI 10.1090/trans2/015/04 - Mikhael Gromov,
*Structures métriques pour les variétés riemanniennes*, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR**682063** - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - Dale Husemoller,
*Fibre bundles*, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR**0229247**, DOI 10.1007/978-1-4757-4008-0 - Nathan Jacobson,
*Basic algebra. II*, W. H. Freeman and Co., San Francisco, Calif., 1980. MR**571884** - Morgan Ward,
*Ring homomorphisms which are also lattice homomorphisms*, Amer. J. Math.**61**(1939), 783–787. MR**10**, DOI 10.2307/2371336 - Saunders MacLane and O. F. G. Schilling,
*Infinite number fields with Noether ideal theories*, Amer. J. Math.**61**(1939), 771–782. MR**19**, DOI 10.2307/2371335 - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - James R. Munkres,
*Topology: a first course*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR**0464128** - Conrad Plaut,
*Geometrizing infinite-dimensional locally compact groups*, Trans. Amer. Math. Soc.**348**(1996), no. 3, 941–962. MR**1348156**, DOI 10.1090/S0002-9947-96-01592-9 - L. S. Pontryagin,
*Topological groups*, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR**0201557** - Neil W. Rickert,
*Arcs in locally compact groups*, Math. Ann.**172**(1967), 222–228. MR**213467**, DOI 10.1007/BF01351189 - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112** - Stallman, C.,
*Dissertation,*University of Tennessee, 1996.

## 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
- 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 - © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1458295