On pairs of metrics invariant under a cocompact action of a group
Author:
S. A. Krat
Journal:
Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 79-86
MSC (2000):
Primary 51K05; Secondary 53C99
DOI:
https://doi.org/10.1090/S1079-6762-01-00097-X
Published electronically:
September 28, 2001
MathSciNet review:
1856889
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Abstract: 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.
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bur Burago, D., Periodic metrics, Representation Theory and Dynamical Systems, pp. 205–210, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992.
buy Buyalo, S. V., Introduction to the metric geometry, St. Petersburg, Obrazovanie, 1997.
gr1 Gromov, M., Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, pp. 79–323, Progr. Math., vol. 144, Birkhäuser, Basel, 1996.
gr2 Gromov, M., Asymptotic invariants of infinite groups, Geometric Group Theory, Vol. 2 (G. A. Noble, M. A. Roller, eds.), London Math. Soc. Lecture Notes Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993.
gr3 Gromov, M., Structures métriques pour les variétés riemanniennes (J. Lafontaine et P. Pansu, eds.), Cedic/Fernand Nathan, Paris, 1981.
krat Krat, S. A., Asymptotic properties of the Heisenberg group, Zap. Nauchn. Seminar. POMI, vol. 261, 1999, pp. 125–154.
l Leichtweiss, K., Konvexe Mengen, Hochschultext, Springer-Verlag, Berlin-New York, 1980.
p Pansu, P., Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), 415–445.
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Additional Information
S. A. Krat
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Email:
krat@math.psu.edu
Keywords:
Metric space,
group action
Received by editor(s):
February 16, 2001
Published electronically:
September 28, 2001
Communicated by:
Richard Schoen
Article copyright:
© Copyright 2001
American Mathematical Society
