A note on elementarity of virtual dendro-morphisms for higher rank lattices

Savini, A.

doi:10.1090/proc/16024

A note on elementarity of virtual dendro-morphisms for higher rank lattices
HTML articles powered by AMS MathViewer

by A. Savini PDF

Proc. Amer. Math. Soc. 150 (2022), 4995-5008 Request permission

Abstract:

Let $\Gamma$ be a discrete countable group and let $(\Omega ,\mu )$ be an ergodic standard Borel probability $\Gamma$-space. Given any non-elementary virtual dendro-morphism (that is a measurable cocycle in the automorphism group of a dendrite), we construct a unitary representation $V$ with no invariant vectors such that $\operatorname {H}^2_b(\Gamma ;V)$ contains a non-zero class. As a consequence, all virtual dendro-morphisms of a higher rank lattice must be elementary.

References

Uri Bader and Alex Furman, Boundaries, rigidity of representations, and Lyapunov exponents, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III, Kyung Moon Sa, Seoul, 2014, pp. 71–96. MR 3729019
U. Bader, A. Furman, and A. Shaker, Superrigidity, weyl groups and actions on the circle, arXiv: 0605276, 2006.
M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 199–235. MR 1694584, DOI 10.1007/s100970050007
M. Burger and N. Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002), no. 2, 219–280. MR 1911660, DOI 10.1007/s00039-002-8245-9
Henk Bruin and Mike Todd, Markov extensions and lifting measures for complex polynomials, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 743–768. MR 2322177, DOI 10.1017/S0143385706000976
Bruno Duchesne and Nicolas Monod, Group actions on dendrites and curves, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2277–2309 (English, with English and French summaries). MR 3893770, DOI 10.5802/aif.3209
Bruno Duchesne and Nicolas Monod, Structural properties of dendrite groups, Trans. AMS 3 (2019), no. 1, 1925–1949.
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
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
Étienne Ghys, Groupes d’homéomorphismes du cercle et cohomologie bornée, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 81–106 (French, with English summary). MR 893858, DOI 10.24033/msmf.358
Étienne Ghys, Actions de réseaux sur le cercle, Invent. Math. 137 (1999), no. 1, 199–231 (French). MR 1703323, DOI 10.1007/s002220050329
Camille Jordan, Sur les assemblages de lignes, J. Reine Angew. Math. 70 (1869), 185–190 (French). MR 1579443, DOI 10.1515/crll.1869.70.185
Casimir Kuratowski, Topologie. Vol. II, Monografie Matematyczne, Tom 21, Państwowe Wydawnictwo Naukowe, Warsaw, 1961 (French). Troisième édition, corrigèe et complétée de deux appendices. MR 0133124
George W. Mackey, Ergodic theory and virtual groups, Math. Ann. 166 (1966), 187–207. MR 201562, DOI 10.1007/BF01361167
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, Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, vol. 1758, Springer-Verlag, Berlin, 2001. MR 1840942, DOI 10.1007/b80626
Nicolas Monod and Yehuda Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom. 67 (2004), no. 3, 395–455. MR 2153026
Sam B. Nadler Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552
Andrés Navas, Reduction of cocycles and groups of diffeomorphisms of the circle, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 2, 193–205. MR 2259900
A. Savini, Parametrized euler class and semicohomology theory, arXiv:2101.11971, 2021.
F. Sarti and A. Savini, Finite reducibility of maximal infinite dimensional measurable cocycles of complex hyperbolic lattices, https://arxiv.org/abs/2005.10529, 2020.
F. Sarti and A. Savini, Superrigidity of maximal measurable cocycles of complex hyperbolic lattices, Math. Z. 300 (2022), no. 1, 421–443. MR 4359531, DOI 10.1007/s00209-021-02801-y
Enhui Shi and Xiangdong Ye, Periodic points for amenable group actions on dendrites, Proc. Amer. Math. Soc. 145 (2017), no. 1, 177–184. MR 3565370, DOI 10.1090/proc/13206
Tadé Ważewski, Sur les courbes de Jordan ne renfermant aucune courbe simple fermée de Jordan, NUMDAM, [place of publication not identified], 1923 (French). MR 3532906
Gordon T. Whyburn, Concerning the Structure of a Continuous Curve, Amer. J. Math. 50 (1928), no. 2, 167–194. MR 1506664, DOI 10.2307/2371754
Gordon T. Whyburn, On the set of all cut points of a continuous curve, Fundam. Math. 15 (1930), no. 1, 185–194.
Dave Witte and Robert J. Zimmer, Actions of semisimple Lie groups on circle bundles, Geom. Dedicata 87 (2001), no. 1-3, 91–121. MR 1866844, DOI 10.1023/A:1012068331112
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