Lie groups that are closed at infinity
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- by Harry F. Hoke PDF
- Trans. Amer. Math. Soc. 313 (1989), 721-735 Request permission
Abstract:
A noncompact Riemannian manifold $M$ is said to be closed at infinity if no bounded volume form which is also bounded away from zero can be written as the exterior derivative of a bounded form on $M$ . The isoperimetric constant of $M$ is defined by $h(M) = \inf \{ {\text {vol}}(\partial S)/{\text {vol}}(S)\}$ where $S$ ranges over compact domains with boundary in $M$. It is shown that a Lie group $G$ with left invariant metric is closed at infinity if and only if $h(G) = 0$ if and only if $G$ is amenable and unimodular. This result relates these geometric invariants of $G$ to the algebraic structure of $G$ since the conditions amenable and unimodular have algebraic characterizations for Lie groups. $G$ is amenable if and only if $G$ is a compact extension of a solvable group and $G$ is unimodular if and only if $\operatorname {Tr}({\text {ad}} X) = 0$ for all $X$ in the Lie algebra of $G$. An application is the clarification of relationships between several conditions for the existence of transversal invariant measures for a foliation of a compact manifold by the orbits of a Lie group action.References
- Robert Brooks, Some Riemannian and dynamical invariants of foliations, Differential geometry (College Park, Md., 1981/1982) Progr. Math., vol. 32, Birkhäuser, Boston, Mass., 1983, pp. 56–72. MR 702527
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
- P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648
- S. E. Goodman and J. F. Plante, Holonomy and averaging in foliated sets, J. Differential Geometry 14 (1979), no. 3, 401–407 (1980). MR 594710
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
- 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
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- Tadeusz Januszkiewicz, Characteristic invariants of noncompact Riemannian manifolds, Topology 23 (1984), no. 3, 289–301. MR 770565, DOI 10.1016/0040-9383(84)90012-0
- J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 232311
- John Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977), no. 2, 178–187. MR 454886, DOI 10.1016/0001-8708(77)90004-4
- M. A. Naĭmark and A. I. Štern, Theory of group representations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 246, Springer-Verlag, New York, 1982. Translated from the Russian by Elizabeth Hewitt; Translation edited by Edwin Hewitt. MR 793377, DOI 10.1007/978-1-4613-8142-6
- J. F. Plante, On the existence of exceptional minimal sets in foliations of codimension one, J. Differential Equations 15 (1974), 178–194. MR 346815, DOI 10.1016/0022-0396(74)90093-X
- J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), no. 2, 327–361. MR 391125, DOI 10.2307/1971034
- Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011
- Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297
- Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 313 (1989), 721-735
- MSC: Primary 58F18; Secondary 22E15
- DOI: https://doi.org/10.1090/S0002-9947-1989-0935533-2
- MathSciNet review: 935533