On the rigidity of Weyl chamber flows and Schur multipliers as topological groups

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2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25

On the rigidity of Weyl chamber flows and Schur multipliers as topological groups

Kurt Vinhage ^1,

1.	Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received June 2014 Revised November 2014 Published May 2015

Abstract
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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.

Keywords: homogeneous flows., cocycles, group actions, partial hyperbolicity, Schur multiplier, Local rigidity.

Mathematics Subject Classification: Primary: 37C85; Secondary: 19C0.

Citation: Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25

References:

[1]	D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 1 (2007), 665-688. doi: 10.3934/jmd.2007.1.665.
[2]	D. Damjanović and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\mathbbR^k$ actions, Discrete Contin. Dyn. Syst., 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985.
[3]	D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbbZ^k$ actions on the torus, Ann. of Math. (2), 172 (2010), 1805-1858. doi: 10.4007/annals.2010.172.1805.
[4]	D. Damjanović and A. Katok, 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, (2011), 4405-4430. doi: 10.1093/imrn/rnq252.
[5]	V. V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: 10.2307/2373853.
[6]	J. L. Dupont, W. Parry and C.-H. Sah, Homology of classical Lie groups made discrete. II. $H_2,H_3,$ and relations with scissors congruences, J. Algebra, 113 (1988), 215-260. doi: 10.1016/0021-8693(88)90191-3.
[7]	A. M. Gleason and R. S. Palais, On a class of transformation groups, Amer. J. Math., 79 (1957), 631-648. doi: 10.2307/2372567.
[8]	M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2), 140 (1994), 295-329. doi: 10.2307/2118602.
[9]	R. Hartshorne, ed., Algebraic Geometry, Corrected reprint of the 1975 original, Proceedings of Symposia in Pure Mathematics, Vol. 29, American Mathematical Society, Providence, R.I., 1979.
[10]	A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova, 216 (1997), Din. Sist. i Smezhnye Vopr., 292-319.
[11]	A. Katok, V. Niţică and A. Török, Non-abelian cohomology of abelian Anosov actions, Ergodic Theory Dynam. Systems, 20 (2000), 259-288. doi: 10.1017/S0143385700000122.
[12]	G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991.
[13]	J. Milnor, Introduction to Algebraic $K$-Theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.
[14]	S. A. Morris, Free products of topological groups, Bull. Austral. Math. Soc., 4 (1971), 17-29. doi: 10.1017/S0004972700046219.
[15]	E. T. Ordman, Free products of topological groups which are $k_{\omega }$-spaces, Trans. Amer. Math. Soc., 191 (1974), 61-73.
[16]	C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity, 13 (1997), 125-179. doi: 10.1006/jcom.1997.0437.
[17]	C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited, J. Mod. Dyn., 6 (2012), 79-120. doi: 10.3934/jmd.2012.6.79.
[18]	C. H. Sah and J. B. Wagoner, Second homology of Lie groups made discrete, Comm. Algebra, 5 (1977), 611-642. 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 (2010), 271-327. 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 (2010), 585-608. doi: 10.3934/jmd.2010.4.585.

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References:

[1]	D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 1 (2007), 665-688. doi: 10.3934/jmd.2007.1.665.
[2]	D. Damjanović and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic $\mathbbR^k$ actions, Discrete Contin. Dyn. Syst., 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985.
[3]	D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbbZ^k$ actions on the torus, Ann. of Math. (2), 172 (2010), 1805-1858. doi: 10.4007/annals.2010.172.1805.
[4]	D. Damjanović and A. Katok, 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, (2011), 4405-4430. doi: 10.1093/imrn/rnq252.
[5]	V. V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: 10.2307/2373853.
[6]	J. L. Dupont, W. Parry and C.-H. Sah, Homology of classical Lie groups made discrete. II. $H_2,H_3,$ and relations with scissors congruences, J. Algebra, 113 (1988), 215-260. doi: 10.1016/0021-8693(88)90191-3.
[7]	A. M. Gleason and R. S. Palais, On a class of transformation groups, Amer. J. Math., 79 (1957), 631-648. doi: 10.2307/2372567.
[8]	M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2), 140 (1994), 295-329. doi: 10.2307/2118602.
[9]	R. Hartshorne, ed., Algebraic Geometry, Corrected reprint of the 1975 original, Proceedings of Symposia in Pure Mathematics, Vol. 29, American Mathematical Society, Providence, R.I., 1979.
[10]	A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova, 216 (1997), Din. Sist. i Smezhnye Vopr., 292-319.
[11]	A. Katok, V. Niţică and A. Török, Non-abelian cohomology of abelian Anosov actions, Ergodic Theory Dynam. Systems, 20 (2000), 259-288. doi: 10.1017/S0143385700000122.
[12]	G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991.
[13]	J. Milnor, Introduction to Algebraic $K$-Theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.
[14]	S. A. Morris, Free products of topological groups, Bull. Austral. Math. Soc., 4 (1971), 17-29. doi: 10.1017/S0004972700046219.
[15]	E. T. Ordman, Free products of topological groups which are $k_{\omega }$-spaces, Trans. Amer. Math. Soc., 191 (1974), 61-73.
[16]	C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity, 13 (1997), 125-179. doi: 10.1006/jcom.1997.0437.
[17]	C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited, J. Mod. Dyn., 6 (2012), 79-120. doi: 10.3934/jmd.2012.6.79.
[18]	C. H. Sah and J. B. Wagoner, Second homology of Lie groups made discrete, Comm. Algebra, 5 (1977), 611-642. 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 (2010), 271-327. 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 (2010), 585-608. doi: 10.3934/jmd.2010.4.585.

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