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Transactions of the American Mathematical Society
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
Erratum: Tran. Amer. Math. Soc. 352 (2000), 5877
MathSciNet review: 1458295
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-99-02086-3
PII: S 0002-9947(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