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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Geometric groups. I
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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.


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:
  • Conrad Plaut
  • Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
  • Email:
  • 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:
  • MathSciNet review: 1458295