## Measurable rigidity of actions on infinite measure homogeneous spaces, II

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- by Alex Furman
- J. Amer. Math. Soc.
**21**(2008), 479-512 - DOI: https://doi.org/10.1090/S0894-0347-07-00588-7
- Published electronically: December 27, 2007
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## Abstract:

We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on*infinite measure*spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups $\Gamma <G$ on homogeneous spaces $G/H$ where $H$ is a sufficiently rich unimodular subgroup in a semi-simple group $G$. We also consider actions of discrete groups of isometries $\Gamma <\mathrm {Isom}(X)$ of a pinched negative curvature space $X$, acting on the space of horospheres $\mathrm {Hor}(X)$. For such systems we prove that the only measurable isomorphisms, joinings, quotients, etc., are the obvious algebraic (or geometric) ones. This work was inspired by the previous work of Shalom and Steger but uses completely different techniques which lead to more general results.

## References

- Martine Babillot and François Ledrappier,
*Geodesic paths and horocycle flow on abelian covers*, Lie groups and ergodic theory (Mumbai, 1996) Tata Inst. Fund. Res. Stud. Math., vol. 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 1–32. MR**1699356**
BFGW Uri Bader, Alex Furman, Alex Gorodnik, and Barak Weiss, - Armand Borel and Jacques Tits,
*Groupes réductifs*, Inst. Hautes Études Sci. Publ. Math.**27**(1965), 55–150 (French). MR**207712** - Armand Borel and Jacques Tits,
*Homomorphismes “abstraits” de groupes algébriques simples*, Ann. of Math. (2)**97**(1973), 499–571 (French). MR**316587**, DOI 10.2307/1970833 - Françoise Dal’bo,
*Remarques sur le spectre des longueurs d’une surface et comptages*, Bol. Soc. Brasil. Mat. (N.S.)**30**(1999), no. 2, 199–221 (French, with English and French summaries). MR**1703039**, DOI 10.1007/BF01235869 - Harry Furstenberg,
*Boundary theory and stochastic processes on homogeneous spaces*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 193–229. MR**0352328** - Harry Furstenberg,
*Rigidity and cocycles for ergodic actions of semisimple Lie groups (after G. A. Margulis and R. Zimmer)*, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 273–292. MR**636529** - Y. Guivarc’h,
*Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés*, Ergodic Theory Dynam. Systems**9**(1989), no. 3, 433–453 (French, with English summary). MR**1016662**, DOI 10.1017/S0143385700005083 - U. Hamenstädt,
*Cocycles, symplectic structures and intersection*, Geom. Funct. Anal.**9**(1999), no. 1, 90–140. MR**1675892**, DOI 10.1007/s000390050082 - Vadim A. Kaimanovich,
*Ergodicity of the horocycle flow*, Dynamical systems (Luminy-Marseille, 1998) World Sci. Publ., River Edge, NJ, 2000, pp. 274–286. MR**1796165** - Vadim A. Kaimanovich,
*SAT actions and ergodic properties of the horosphere foliation*, Rigidity in dynamics and geometry (Cambridge, 2000) Springer, Berlin, 2002, pp. 261–282. MR**1919405** - Alexander Lubotzky,
*Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem*, Ann. of Math. (2)**144**(1996), no. 2, 441–452. MR**1418904**, DOI 10.2307/2118597 - G. A. Margulis,
*Discrete subgroups of semisimple Lie groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR**1090825**, DOI 10.1007/978-3-642-51445-6 - Jean-Pierre Otal,
*Le spectre marqué des longueurs des surfaces à courbure négative*, Ann. of Math. (2)**131**(1990), no. 1, 151–162 (French). MR**1038361**, DOI 10.2307/1971511 - Jean-Pierre Otal,
*The hyperbolization theorem for fibered 3-manifolds*, SMF/AMS Texts and Monographs, vol. 7, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1996 French original by Leslie D. Kay. MR**1855976** - M. S. Raghunathan and T. N. Venkataramana,
*The first Betti number of arithmetic groups and the congruence subgroup problem*, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 95–107. MR**1247500**, DOI 10.1090/conm/153/01308 - 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,
*Interactions between ergodic theory, Lie groups, and number theory*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 157–182. MR**1403920**
ShalomSteger:HomSpI Yehuda Shalom and Tim Steger, - Dennis Sullivan,
*Discrete conformal groups and measurable dynamics*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 1, 57–73. MR**634434**, DOI 10.1090/S0273-0979-1982-14966-7 - Chengbo Yue,
*The ergodic theory of discrete isometry groups on manifolds of variable negative curvature*, Trans. Amer. Math. Soc.**348**(1996), no. 12, 4965–5005. MR**1348871**, DOI 10.1090/S0002-9947-96-01614-5

*Rigidity of group actions on homogeneous spaces, III*, 2007, in preparation.

*Rigidity of group actions on infinite volume homogeneous spaces I*, unpublished.

## Bibliographic Information

**Alex Furman**- Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
- Email: furman@math.uic.edu
- Received by editor(s): March 8, 2006
- Published electronically: December 27, 2007
- Additional Notes: The author was supported in part by NSF grant DMS-0094245 and BSF USA-Israel grant 2004345.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**21**(2008), 479-512 - MSC (2000): Primary 37A17; Secondary 37A35, 22E40, 22F30
- DOI: https://doi.org/10.1090/S0894-0347-07-00588-7
- MathSciNet review: 2373357