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Consider two intrinsic metrics invariant under the same cocompact action of an abelian group. Assume that the ratio of the distances tends to one as the distances grow to infinity. Then it is known (due to D. Burago) that the difference between the metric functions is uniformly bounded. We will prove an analog of this result for hyperbolic groups, as well as a partial generalization of this result for the Heisenberg group: a word metric on the Heisenberg group lies within bounded GH distance from its asymptotic cone.

Copyright 2001 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 07 (2001), pp. 79-86
• Publisher Identifier: S 1079-6762(01)00097-X
• 2000 Mathematics Subject Classification. Primary 51K05; Secondary 53C99
• Key words and phrases. Metric space, group action
• Received by the editors February 16, 2001
• Posted on September 28, 2001
• Communicated by Richard Schoen
• Comments (When Available)

S. A. Krat

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

E-mail address: krat@math.psu.edu