Book Review
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4321455
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Book Information:
Author:
Robert J. Zimmer
Title:
Group actions in ergodic theory, geometry, and topology: Selected papers
Additional book information:
University of Chicago Press,
Chicago, IL,
2020,
672 pp.,
ISBN 978-0-226-56827-0
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
A. Brown, D. Fisher, and S. Hurtado, Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T), arXiv:1608.04995 (2016).
Aaron Brown, David Fisher, and Sebastian Hurtado, Zimmer’s conjecture for actions of $\textrm {SL}(m, \Bbb Z)$, Invent. Math. 221 (2020), no. 3, 1001–1060. MR 4132960, DOI 10.1007/s00222-020-00962-x
Aaron Brown, Federico Rodriguez Hertz, and Zhiren Wang, Global smooth and topological rigidity of hyperbolic lattice actions, Ann. of Math. (2) 186 (2017), no. 3, 913–972. MR 3702679, DOI 10.4007/annals.2017.186.3.3
Eugenio Calabi, On compact, Riemannian manifolds with constant curvature. I, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 155–180. MR 0133787
Eugenio Calabi and Edoardo Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 (1960), 472–507. MR 111058, DOI 10.2307/1969939
David Fisher, Boris Kalinin, and Ralf Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds, J. Amer. Math. Soc. 26 (2013), no. 1, 167–198. With an appendix by James F. Davis. MR 2983009, DOI 10.1090/S0894-0347-2012-00751-6
Alex Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999), no. 3, 1083–1108. MR 1740985, DOI 10.2307/121063
Alex Furman, A survey of measured group theory, Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 296–374. MR 2807836
Alex Furman and Nicolas Monod, Product groups acting on manifolds, Duke Math. J. 148 (2009), no. 1, 1–39. MR 2515098, DOI 10.1215/00127094-2009-018
Harry Furstenberg, Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc. 73 (1967), 350–356. MR 210812, DOI 10.1090/S0002-9904-1967-11748-8
Damien Gaboriau, Orbit equivalence and measured group theory, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1501–1527. MR 2827853
Adrian Ioana, Sorin Popa, and Stefaan Vaes, A class of superrigid group von Neumann algebras, Ann. of Math. (2) 178 (2013), no. 1, 231–286. MR 3043581, DOI 10.4007/annals.2013.178.1.4
A. Jackson, Interview with A. Connes, Celebratio Mathematica, 2021.
Anatole Katok and Ralf J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 131–156. MR 1307298
G. A. Margulis, Factor groups of discrete subgroups and measure theory, Funktsional. Anal. i Prilozhen. 12 (1978), no. 4, 64–76 (Russian). MR 515630
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
Nicolas Monod and Yehuda Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006), no. 3, 825–878. MR 2259246, DOI 10.4007/annals.2006.164.825
G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383
Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753, DOI 10.1090/S0273-0979-1980-14702-3
Sorin Popa, Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups, Invent. Math. 170 (2007), no. 2, 243–295. MR 2342637, DOI 10.1007/s00222-007-0063-0
Sorin Popa, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008), no. 4, 981–1000. MR 2425177, DOI 10.1090/S0894-0347-07-00578-4
Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324
Yehuda Shalom, Measurable group theory, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 391–423. MR 2185757
André Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384. MR 137792, DOI 10.2307/1970140
André Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578–602. MR 137793, DOI 10.2307/1970212
Robert J. Zimmer, Amenable ergodic actions, hyperfinite factors, and Poincaré flows, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1078–1080. MR 460598, DOI 10.1090/S0002-9904-1977-14392-9
Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis 27 (1978), no. 3, 350–372. MR 0473096, DOI 10.1016/0022-1236(78)90013-7
Robert J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2) 112 (1980), no. 3, 511–529. MR 595205, DOI 10.2307/1971090
Robert J. Zimmer, Volume preserving actions of lattices in semisimple groups on compact manifolds, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 5–33. MR 743815
R. J. Zimmer, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), no. 3, 425–436. MR 735334, DOI 10.1007/BF01388637
Robert J. Zimmer, Lattices in semisimple groups and distal geometric structures, Invent. Math. 80 (1985), no. 1, 123–137. MR 784532, DOI 10.1007/BF01388551
Robert J. Zimmer, Spectrum, entropy, and geometric structures for smooth actions of Kazhdan groups, Israel J. Math. 75 (1991), no. 1, 65–80. MR 1147291, DOI 10.1007/BF02787182
Robert J. Zimmer, Entropy and arithmetic quotients for simple automorphism groups of geometric manifolds, Geom. Dedicata 107 (2004), 47–56. MR 2110753, DOI 10.1023/B:GEOM.0000049088.17279.57
References
- D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
- A. Brown, D. Fisher, and S. Hurtado, Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T), arXiv:1608.04995 (2016).
- Aaron Brown, David Fisher, and Sebastian Hurtado, Zimmer’s conjecture for actions of $\mathrm {SL}(m, \mathbb {Z})$, Invent. Math. 221 (2020), no. 3, 1001–1060. MR 4132960, DOI 10.1007/s00222-020-00962-x
- Aaron Brown, Federico Rodriguez Hertz, and Zhiren Wang, Global smooth and topological rigidity of hyperbolic lattice actions, Ann. of Math. (2) 186 (2017), no. 3, 913–972. MR 3702679, DOI 10.4007/annals.2017.186.3.3
- Eugenio Calabi, On compact, Riemannian manifolds with constant curvature. I, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 155–180. MR 0133787
- Eugenio Calabi and Edoardo Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 (1960), 472–507. MR 111058, DOI 10.2307/1969939
- David Fisher, Boris Kalinin, and Ralf Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds, J. Amer. Math. Soc. 26 (2013), no. 1, 167–198. With an appendix by James F. Davis. MR 2983009, DOI 10.1090/S0894-0347-2012-00751-6
- Alex Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999), no. 3, 1083–1108. MR 1740985, DOI 10.2307/121063
- Alex Furman, A survey of measured group theory, Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 296–374. MR 2807836
- Alex Furman and Nicolas Monod, Product groups acting on manifolds, Duke Math. J. 148 (2009), no. 1, 1–39. MR 2515098, DOI 10.1215/00127094-2009-018
- Harry Furstenberg, Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc. 73 (1967), 350–356. MR 210812, DOI 10.1090/S0002-9904-1967-11748-8
- Damien Gaboriau, Orbit equivalence and measured group theory, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1501–1527. MR 2827853
- Adrian Ioana, Sorin Popa, and Stefaan Vaes, A class of superrigid group von Neumann algebras, Ann. of Math. (2) 178 (2013), no. 1, 231–286. MR 3043581, DOI 10.4007/annals.2013.178.1.4
- A. Jackson, Interview with A. Connes, Celebratio Mathematica, 2021.
- Anatole Katok and Ralf J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 131–156. MR 1307298
- G. A. Margulis, Factor groups of discrete subgroups and measure theory, Funktsional. Anal. i Prilozhen. 12 (1978), no. 4, 64–76 (Russian). MR 515630
- 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
- Nicolas Monod and Yehuda Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006), no. 3, 825–878. MR 2259246, DOI 10.4007/annals.2006.164.825
- G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383
- Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753, DOI 10.1090/S0273-0979-1980-14702-3
- Sorin Popa, Cocycle and orbit equivalence superrigidity for malleable actions of $w$-rigid groups, Invent. Math. 170 (2007), no. 2, 243–295. MR 2342637, DOI 10.1007/s00222-007-0063-0
- Sorin Popa, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008), no. 4, 981–1000. MR 2425177, DOI 10.1090/S0894-0347-07-00578-4
- Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324
- Yehuda Shalom, Measurable group theory, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 391–423. MR 2185757
- André Weil, On discrete subgroups of Lie groups, Ann. of Math. (2) 72 (1960), 369–384. MR 137792, DOI 10.2307/1970140
- André Weil, On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578–602. MR 137793, DOI 10.2307/1970212
- Robert J. Zimmer, Amenable ergodic actions, hyperfinite factors, and Poincaré flows, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1078–1080. MR 460598, DOI 10.1090/S0002-9904-1977-14392-9
- Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis 27 (1978), no. 3, 350–372. MR 0473096, DOI 10.1016/0022-1236(78)90013-7
- Robert J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2) 112 (1980), no. 3, 511–529. MR 595205, DOI 10.2307/1971090
- Robert J. Zimmer, Volume preserving actions of lattices in semisimple groups on compact manifolds, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 5–33. MR 743815
- R. J. Zimmer, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), no. 3, 425–436. MR 735334, DOI 10.1007/BF01388637
- Robert J. Zimmer, Lattices in semisimple groups and distal geometric structures, Invent. Math. 80 (1985), no. 1, 123–137. MR 784532, DOI 10.1007/BF01388551
- Robert J. Zimmer, Spectrum, entropy, and geometric structures for smooth actions of Kazhdan groups, Israel J. Math. 75 (1991), no. 1, 65–80. MR 1147291, DOI 10.1007/BF02787182
- Robert J. Zimmer, Entropy and arithmetic quotients for simple automorphism groups of geometric manifolds, Geom. Dedicata 107 (2004), 47–56. MR 2110753, DOI 10.1023/B:GEOM.0000049088.17279.57
Review Information:
Reviewer:
David Kerr
Affiliation:
Mathematisches Institut, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany
Journal:
Bull. Amer. Math. Soc.
58 (2021), 627-634
DOI:
https://doi.org/10.1090/bull/1741
Published electronically:
July 6, 2021
Review copyright:
© Copyright 2021
American Mathematical Society
