Geometric groups. I |
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Authors: | Valera Berestovskii Conrad Plaut Cornelius Stallman |
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Institution: | Department of Mathematics, Omsk State University, Pr. Mira 55A, Omsk 77 644077 Russia ; Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 ; Department of Mathematics and Computer Science, Augusta State University, Augusta, Georgia 30904-2200 |
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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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Keywords: | Topological groups Lie groups invariant metrics Alexandrov curvature bounded below universal covering group |
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