Vinhage, Kurt On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. (English) Zbl 1358.37060 J. Mod. Dyn. 9, 25-49 (2015). Summary: We effectively conclude the local rigidity program for generic restrictions of partially hyperbolic Weyl chamber flows. Our methods replace and extend previous ones by circumventing computations made in Schur multipliers. Instead, we construct a natural topology on \(H_2(G,\mathbb{Z})\), and rely on classical Lie structure theory for central extensions. Cited in 3 Documents MSC: 37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) 19C09 Central extensions and Schur multipliers 22E40 Discrete subgroups of Lie groups 22E46 Semisimple Lie groups and their representations 37D30 Partially hyperbolic systems and dominated splittings Keywords:local rigidity; group actions; Schur multiplier; cocycles; partial hyperbolicity; homogeneous flows PDFBibTeX XMLCite \textit{K. Vinhage}, J. Mod. Dyn. 9, 25--49 (2015; Zbl 1358.37060) Full Text: DOI arXiv References: [1] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Mod. Dyn., 1, 665 (2007) · Zbl 1139.37017 · doi:10.3934/jmd.2007.1.665 [2] D. Damjanović, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic \(\mathbbR^k\) actions,, Discrete Contin. Dyn. Syst., 13, 985 (2005) · Zbl 1109.37029 · doi:10.3934/dcds.2005.13.985 [3] D. Damjanović, Local rigidity of partially hyperbolic actions I. KAM method and \(\mathbbZ^k\) actions on the torus,, Ann. of Math. (2), 172, 1805 (2010) · Zbl 1209.37017 · doi:10.4007/annals.2010.172.1805 [4] D. Damjanović, Local rigidity of partially hyperbolic actions. II: The geometric method and restrictions of Weyl chamber flows on \(SL \[(n,\mathbbR)\]/\Gamma \),, Int. Math. Res. Not. IMRN, 4405 (2011) · Zbl 1291.37027 · doi:10.1093/imrn/rnq252 [5] V. V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100, 303 (1978) · Zbl 0392.20027 · doi:10.2307/2373853 [6] J. L. Dupont, Homology of classical Lie groups made discrete. II. \(H_2,H_3,\) and relations with scissors congruences,, J. Algebra, 113, 215 (1988) · Zbl 0657.55022 · doi:10.1016/0021-8693(88)90191-3 [7] A. M. Gleason, On a class of transformation groups,, Amer. J. Math., 79, 631 (1957) · Zbl 0084.03203 · doi:10.2307/2372567 [8] M. Grayson, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140, 295 (1994) · Zbl 0824.58032 · doi:10.2307/2118602 [9] R. Hartshorne, <em>Algebraic Geometry</em>,, Corrected reprint of the 1975 original (1975) · Zbl 0532.14001 [10] A. Katok, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Tr. Mat. Inst. Steklova, 216, 292 (1997) · Zbl 0938.37010 [11] A. Katok, Non-abelian cohomology of abelian Anosov actions,, Ergodic Theory Dynam. Systems, 20, 259 (2000) · Zbl 0977.57042 · doi:10.1017/S0143385700000122 [12] G. A. Margulis, <em>Discrete Subgroups of Semisimple Lie Groups</em>,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (1991) · Zbl 0732.22008 [13] J. Milnor, <em>Introduction to Algebraic \(K\)-Theory,</em>, Annals of Mathematics Studies (1971) · Zbl 0237.18005 [14] S. A. Morris, Free products of topological groups,, Bull. Austral. Math. Soc., 4, 17 (1971) · Zbl 0199.34503 · doi:10.1017/S0004972700046219 [15] E. T. Ordman, Free products of topological groups which are \(k_{\omega }\)-spaces,, Trans. Amer. Math. Soc., 191, 61 (1974) · Zbl 0287.22003 [16] C. Pugh, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity, 13, 125 (1997) · Zbl 0883.58025 · doi:10.1006/jcom.1997.0437 [17] C. Pugh, Hölder foliations, revisited,, J. Mod. Dyn., 6, 79 (2012) · Zbl 1259.37024 · doi:10.3934/jmd.2012.6.79 [18] C. H. Sah, Second homology of Lie groups made discrete,, Comm. Algebra, 5, 611 (1977) · Zbl 0375.18006 · doi:10.1080/00927877708822184 [19] Z. Wang, Local rigidity of partially hyperbolic actions: Twisted symmetric space examples,, preprint. [20] Z. J. Wang, Local rigidity of partially hyperbolic actions,, J. Mod. Dyn., 4, 271 (2010) · Zbl 1205.37048 · doi:10.3934/jmd.2010.4.271 [21] Z. J. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions,, J. Mod. Dyn., 4, 585 (2010) · Zbl 1223.37037 · doi:10.3934/jmd.2010.4.585 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.