Measurable quotients of unipotent translations on homogeneous spaces

Witte, Dave

doi:10.1090/S0002-9947-1994-1181187-4

Measurable quotients of unipotent translations on homogeneous spaces
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by Dave Witte

Trans. Amer. Math. Soc. 345 (1994), 577-594

DOI: https://doi.org/10.1090/S0002-9947-1994-1181187-4

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Correction: Trans. Amer. Math. Soc. 349 (1997), 4685-4688.

Abstract:

Let U be a nilpotent, unipotent subgroup of a Lie group G, and let $\Gamma$ be a closed subgroup of G. Marina Ratner showed that every ergodic U-invariant probability measure on the homogeneous space $\Gamma \backslash G$ is of a simple algebraic form. We use this fundamental new result to show that every measurable quotient of the U-action on $\Gamma \backslash G$ is of a simple algebraic form. Roughly speaking, any quotient is a double-coset space $\Lambda \backslash G/K$.

References

L. Auslander and J. Brezin, Almost algebraic Lie algebras, J. Algebra 8 (1968), 295–313. MR 224745, DOI 10.1016/0021-8693(68)90061-6
Nicolas Bourbaki, Éléments de mathématique: groupes et algèbres de Lie, Masson, Paris, 1982 (French). Chapitre 9. Groupes de Lie réels compacts. [Chapter 9. Compact real Lie groups]. MR 682756
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
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
A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), no. 4, 531–557. MR 922364, DOI 10.1017/S0143385700004193
Gerhard P. Hochschild, Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, vol. 75, Springer-Verlag, New York-Berlin, 1981. MR 620024
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
Calvin C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154–178. MR 193188, DOI 10.2307/2373052
Calvin C. Moore, The Mautner phenomenon for general unitary representations, Pacific J. Math. 86 (1980), no. 1, 155–169. MR 586875
G. D. Mostow, On the fundamental group of a homogeneous space, Ann. of Math. (2) 66 (1957), 249–255. MR 88675, DOI 10.2307/1969997

action on

William Parry, Metric classification of ergodic nilflows and unipotent affines, Amer. J. Math. 93 (1971), 819–828. MR 284567, DOI 10.2307/2373472
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
Marina Ratner, Rigidity of horocycle flows, Ann. of Math. (2) 115 (1982), no. 3, 597–614. MR 657240, DOI 10.2307/2007014
Marina Ratner, Factors of horocycle flows, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 465–489 (1983). MR 721735, DOI 10.1017/S0143385700001723
Marina Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. (2) 118 (1983), no. 2, 277–313. MR 717825, DOI 10.2307/2007030
Marina Ratner, Ergodic theory in hyperbolic space, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 309–334. MR 737411, DOI 10.1090/conm/026/737411
Marina Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), no. 2, 449–482. MR 1062971, DOI 10.1007/BF01231511
Marina Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), no. 3-4, 229–309. MR 1075042, DOI 10.1007/BF02391906
Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, DOI 10.2307/2944357
V. A. Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949), 107–150 (Russian). MR 0030584
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
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
Dave Witte, Zero-entropy affine maps on homogeneous spaces, Amer. J. Math. 109 (1987), no. 5, 927–961. MR 910358, DOI 10.2307/2374495
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4