American Institute of Mathematical Sciences

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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

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.  doi: 10.3934/jmd.2007.1.665.  Google Scholar

[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.  doi: 10.3934/dcds.2005.13.985.  Google Scholar

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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.  doi: 10.4007/annals.2010.172.1805.  Google Scholar

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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.  doi: 10.1093/imrn/rnq252.  Google Scholar

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V. V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100 (1978), 303.  doi: 10.2307/2373853.  Google Scholar

[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.  doi: 10.1016/0021-8693(88)90191-3.  Google Scholar

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A. M. Gleason and R. S. Palais, On a class of transformation groups,, Amer. J. Math., 79 (1957), 631.  doi: 10.2307/2372567.  Google Scholar

[8]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140 (1994), 295.  doi: 10.2307/2118602.  Google Scholar

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R. Hartshorne, ed., Algebraic Geometry,, Corrected reprint of the 1975 original, (1975).   Google Scholar

[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), 292.   Google Scholar

[11]

A. Katok, V. Niţică and A. Török, Non-abelian cohomology of abelian Anosov actions,, Ergodic Theory Dynam. Systems, 20 (2000), 259.  doi: 10.1017/S0143385700000122.  Google Scholar

[12]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1991).   Google Scholar

[13]

J. Milnor, Introduction to Algebraic $K$-Theory,, Annals of Mathematics Studies, (1971).   Google Scholar

[14]

S. A. Morris, Free products of topological groups,, Bull. Austral. Math. Soc., 4 (1971), 17.  doi: 10.1017/S0004972700046219.  Google Scholar

[15]

E. T. Ordman, Free products of topological groups which are $k_{\omega }$-spaces,, Trans. Amer. Math. Soc., 191 (1974), 61.   Google Scholar

[16]

C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity, 13 (1997), 125.  doi: 10.1006/jcom.1997.0437.  Google Scholar

[17]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited,, J. Mod. Dyn., 6 (2012), 79.  doi: 10.3934/jmd.2012.6.79.  Google Scholar

[18]

C. H. Sah and J. B. Wagoner, Second homology of Lie groups made discrete,, Comm. Algebra, 5 (1977), 611.  doi: 10.1080/00927877708822184.  Google Scholar

[19]

Z. Wang, Local rigidity of partially hyperbolic actions: Twisted symmetric space examples,, preprint., ().   Google Scholar

[20]

Z. J. Wang, Local rigidity of partially hyperbolic actions,, J. Mod. Dyn., 4 (2010), 271.  doi: 10.3934/jmd.2010.4.271.  Google Scholar

[21]

Z. J. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions,, J. Mod. Dyn., 4 (2010), 585.  doi: 10.3934/jmd.2010.4.585.  Google Scholar

show all references

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.  doi: 10.3934/jmd.2007.1.665.  Google Scholar

[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.  doi: 10.3934/dcds.2005.13.985.  Google Scholar

[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.  doi: 10.4007/annals.2010.172.1805.  Google Scholar

[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.  doi: 10.1093/imrn/rnq252.  Google Scholar

[5]

V. V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100 (1978), 303.  doi: 10.2307/2373853.  Google Scholar

[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.  doi: 10.1016/0021-8693(88)90191-3.  Google Scholar

[7]

A. M. Gleason and R. S. Palais, On a class of transformation groups,, Amer. J. Math., 79 (1957), 631.  doi: 10.2307/2372567.  Google Scholar

[8]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms,, Ann. of Math. (2), 140 (1994), 295.  doi: 10.2307/2118602.  Google Scholar

[9]

R. Hartshorne, ed., Algebraic Geometry,, Corrected reprint of the 1975 original, (1975).   Google Scholar

[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), 292.   Google Scholar

[11]

A. Katok, V. Niţică and A. Török, Non-abelian cohomology of abelian Anosov actions,, Ergodic Theory Dynam. Systems, 20 (2000), 259.  doi: 10.1017/S0143385700000122.  Google Scholar

[12]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1991).   Google Scholar

[13]

J. Milnor, Introduction to Algebraic $K$-Theory,, Annals of Mathematics Studies, (1971).   Google Scholar

[14]

S. A. Morris, Free products of topological groups,, Bull. Austral. Math. Soc., 4 (1971), 17.  doi: 10.1017/S0004972700046219.  Google Scholar

[15]

E. T. Ordman, Free products of topological groups which are $k_{\omega }$-spaces,, Trans. Amer. Math. Soc., 191 (1974), 61.   Google Scholar

[16]

C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity,, J. Complexity, 13 (1997), 125.  doi: 10.1006/jcom.1997.0437.  Google Scholar

[17]

C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited,, J. Mod. Dyn., 6 (2012), 79.  doi: 10.3934/jmd.2012.6.79.  Google Scholar

[18]

C. H. Sah and J. B. Wagoner, Second homology of Lie groups made discrete,, Comm. Algebra, 5 (1977), 611.  doi: 10.1080/00927877708822184.  Google Scholar

[19]

Z. Wang, Local rigidity of partially hyperbolic actions: Twisted symmetric space examples,, preprint., ().   Google Scholar

[20]

Z. J. Wang, Local rigidity of partially hyperbolic actions,, J. Mod. Dyn., 4 (2010), 271.  doi: 10.3934/jmd.2010.4.271.  Google Scholar

[21]

Z. J. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions,, J. Mod. Dyn., 4 (2010), 585.  doi: 10.3934/jmd.2010.4.585.  Google Scholar

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