Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T02:38:56.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Cones and Ultrapowers of Lie Groups

Published online by Cambridge University Press:  15 January 2014

Linus Kramer  and
Katrin Tent
Linus Kramer
Affiliation:
Fachbereich Mathematik, Tu Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, GermanyE-mail: [email protected]
Katrin Tent
Affiliation:
Mathematisches Institut, Universität WürzburgAm Hubland D-97074 Würzburg, GermanyE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

§1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.

§2. Definitions. The intuitive idea behind Gromov's concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/n for n ϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromov's original definition, the asymptotic cone existed only for a rather restricted class of spaces.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 2 , June 2004 , pp. 175 - 185
Copyright
Copyright © Association for Symbolic Logic 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Alperin, R. and Bass, H., Length functions of group actions on ∧-trees, Combinatorial group theory and topology (Alta, Utah, 1984), Princeton University Press, Princeton, NJ, 1987, pp. 265–378.CrossRefGoogle Scholar
[2] Bennett, C., Affine ∧-buildings. I, Proceedings of the London Mathematical Society., vol. 68 (1994), pp. 541–576.Google Scholar
[3] Bridson, M. and Haefliger, A., Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999.Google Scholar
[4] Brown, K., Buildings, Springer-Verlag, New York, 1989.CrossRefGoogle Scholar
[5] Chang, C. C. and Keisler, H. J., Model theory, 3rd ed., North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
[6] de la Harpe, P., Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.Google Scholar
[7] Eberlein, P., Geometry of nonpositively curved manifolds, University of Chicago Press, Chicago, IL, 1996.Google Scholar
[8] Gromov, M., Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991), Cambridge University Press, Cambridge, 1993, pp. 1–295.Google Scholar
[9] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Academic Press Inc., New York-London, 1978.Google Scholar
[10] Kleiner, B. and Leeb, B., Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Institut des Hautes Études Scientifiques. Publications Mathématiques, (1997), no. 86, pp. 115–197.Google Scholar
[11] Kramer, L., Snelah, S., Tent, K., and Thomas, S., Asymptotic cones of finitely presented groups , submitted.Google Scholar
[12] Kramer, L. and Tent, K., Affine ∧-buildings and the Margulis' conjecture , inpreparation.Google Scholar
[13] Lightstone, A. H. and Robmson, A., Nonarchimedean fields and asymptotic expansions, North-Holland Publishing Co., Amsterdam, 1975.Google Scholar
[14] Luxemburg, W., On a class of valuation fields introduced by A. Robinson, Israel Journal of Mathematics, vol. 25 (1976), pp. 189–201.CrossRefGoogle Scholar
[15] Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Annals of Mathematics, vol. 129 (1989), pp. 1–60.Google Scholar
[16] Parreau, A., Dégénérescences de sous-groupes discrets de groupes de Lie semisimples et actions de groupes sur des immeubles affines, These , Université de Paris-Sud XI Orsay, 2000.Google Scholar
[17] Point, F., Groups of polynomial growth and their associated metric spaces, Journal of Algebra, vol. 175 (1995), no. 1, pp. 105–121.CrossRefGoogle Scholar
[18] Roitman, J., Nonisomorphic hyper-real fields from nonisomorphic ultrapowers, Mathematische Zeitschrift, vol. 181 (1982), no. 1, pp. 93–96.CrossRefGoogle Scholar
[19] Ronan, M., Lectures on buildings, Academic Press Inc., Boston, MA, 1989.Google Scholar
[20] Thomas, S. and Vellckovic, B., Asymptotic cones of finitely generated groups, The Bulletin of the London Mathematical Society, vol. 32 (2000), no. 2, pp. 203–208.Google Scholar
[21] Thornton, B., Asymptotic cones of symmetric spaces, Ph.D. thesis , University of Utah, 2002.Google Scholar
[22] Tits, J., Buildings of spherical type and finite BN-pairs, Springer-Verlag, Berlin, 1974.Google Scholar
[23] Tits, J., Immeubles de type affine, Buildings and the geometry of diagrams (Como, 1984), Lecture Notes in Mathematics, vol. 1181, Springer, Berlin, 1986, pp. 159–190.CrossRefGoogle Scholar
[24] van den Dries, L. and Wilkie, A., Gromov's theorem on groups of polynomial growth and elementary logic, Journal of Algebra, vol. 89 (1984), pp. 349–374.Google Scholar