Tessellations of solvmanifolds
HTML articles powered by AMS MathViewer
- by Dave Witte
- Trans. Amer. Math. Soc. 350 (1998), 3767-3796
- DOI: https://doi.org/10.1090/S0002-9947-98-01980-1
- PDF | Request permission
Abstract:
Let $A$ be a closed subgroup of a connected, solvable Lie group $G$, such that the homogeneous space $A\backslash G$ is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation $A\backslash G/\Gamma$ of $A\backslash G$ is finitely covered by a compact homogeneous space $G’/\Gamma ’$. We prove that the covering map can be taken to be very well behaved — a “crossed" affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on $A\backslash G/\Gamma$ that has a dense orbit is covered by a natural flow on $G’/\Gamma ’$.References
- Louis Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. MR 486307, DOI 10.1090/S0002-9904-1973-13134-9
- Louis Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227–261. MR 486307, DOI 10.1090/S0002-9904-1973-13134-9
- L. Auslander, L. Green, and F. Hahn, Flows on homogeneous spaces, Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 1963. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. MR 0167569, DOI 10.1515/9781400882021
- L. Auslander and R. Tolimieri, Splitting theorems and the structure of solvmanifolds, Ann. of Math. (2) 92 (1970), 164–173. MR 276995, DOI 10.2307/1970700
- James S. DeGracie and Wayne A. Fuller, Estimation of the slope and analysis of covariance when the concomitant variable is measured with error, J. Amer. Statist. Assoc. 67 (1972), 930–937. MR 365930, DOI 10.1080/01621459.1972.10481321
- Armand Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179–188. MR 123639, DOI 10.2307/1970150
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
- Jonathan Brezin and Calvin C. Moore, Flows on homogeneous spaces: a new look, Amer. J. Math. 103 (1981), no. 3, 571–613. MR 618325, DOI 10.2307/2374105
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Leonard S. Charlap, Bieberbach groups and flat manifolds, Universitext, Springer-Verlag, New York, 1986. MR 862114, DOI 10.1007/978-1-4613-8687-2
- P. B. Chen and T. S. Wu, On full subgroups of solvable groups, Amer. J. Math. 108 (1986), no. 6, 1487–1506. MR 868900, DOI 10.2307/2374534
- S. G. Dani, On ergodic quasi-invariant measures of group automorphism, Israel J. Math. 43 (1982), no. 1, 62–74. MR 728879, DOI 10.1007/BF02761685
- J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
- F. Thomas Farrell and L. Edwin Jones, Classical aspherical manifolds, CBMS Regional Conference Series in Mathematics, vol. 75, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1056079, DOI 10.1090/cbms/075
- David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI 10.1016/0001-8708(83)90053-1
- Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
- V. I. Istrăţescu, On a class of measures on locally compact Abelian groups, Rev. Roumaine Math. Pures Appl. 11 (1966), 431–434. MR 201918
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- José María Montesinos, Classical tessellations and three-manifolds, Universitext, Springer-Verlag, Berlin, 1987. MR 915761, DOI 10.1007/978-3-642-61572-6
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- William Parry, Dynamical representations in nilmanifolds, Compositio Math. 26 (1973), 159–174. MR 320277
- M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
- C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
- Dave Witte, Zero-entropy affine maps on homogeneous spaces, Amer. J. Math. 109 (1987), no. 5, 927–961. MR 910358, DOI 10.2307/2374495
- Dave Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math. 122 (1995), no. 1, 147–193. MR 1354957, DOI 10.1007/BF01231442
- Jacques Tits, Le Monstre (d’après R. Griess, B. Fischer et al.), Astérisque 121-122 (1985), 105–122 (French). Seminar Bourbaki, Vol. 1983/84. MR 768956
Bibliographic Information
- Dave Witte
- Affiliation: Department of Mathematics, Williams College, Williamstown, MA 01267
- Address at time of publication: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- Email: dwitte@math.okstate.edu
- Received by editor(s): October 6, 1994
- Received by editor(s) in revised form: November 5, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3767-3796
- MSC (1991): Primary 22E25, 22E40, 53C30; Secondary 05B45, 20G20, 20H15, 57S20, 57S30
- DOI: https://doi.org/10.1090/S0002-9947-98-01980-1
- MathSciNet review: 1432206