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ISSN 1079-6762

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
Published electronically: September 28, 2001
MathSciNet review: 1856889
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Abstract | References | Similar Articles | Additional Information

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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  • 1. V. N. Berestovskiĭ, Geodesics of the left-invariant nonholonomic Riemannian metric on the group of motions in the Euclidean plane, Sibirsk. Mat. Zh. 35 (1994), no. 6, 1223–1229, i (Russian, with Russian summary); English transl., Siberian Math. J. 35 (1994), no. 6, 1083–1088. MR 1317534 (95m:58034), http://dx.doi.org/10.1007/BF02104709
  • 2. D. Yu. Burago, Periodic metrics, Representation theory and dynamical systems, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 205–210. MR 1166203 (93c:53029)
  • 3. Buyalo, S. V., Introduction to the metric geometry, St. Petersburg, Obrazovanie, 1997.
  • 4. Mikhael Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR 1421823 (2000f:53034)
  • 5. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544 (95m:20041)
  • 6. Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063 (85e:53051)
  • 7. Krat, S. A., Asymptotic properties of the Heisenberg group, Zap. Nauchn. Seminar. POMI, vol. 261, 1999, pp. 125-154. CMP 2000:12
  • 8. K. Leichtweiss, Konvexe Mengen, Hochschulbücher für Mathematik [University Books for Mathematics], 81, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980 (German). MR 559138 (81b:52001)
  • 9. Pierre Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 415–445 (French, with English summary). MR 741395 (85m:53040), http://dx.doi.org/10.1017/S0143385700002054

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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

DOI: http://dx.doi.org/10.1090/S1079-6762-01-00097-X
PII: S 1079-6762(01)00097-X
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