Rigidity of certain solvable actions on the sphere

Asaoka, Masayuki¹

¹ Department of Mathematics, Kyoto University, Kita-shirakawa Oiwake-cho, Sakyo, Kyoto 606-8502, Japan

Geometry & topology, Tome 16 (2012) no. 3, pp. 1835-1857.

Résumé

An analog of the Baumslag–Solitar group $BS\left(1,k\right)$ naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimensions, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure.

Détail

DOI : 10.2140/gt.2012.16.1835

Classification : 37C85
Keywords: group actions, local rigidity

Bibliographie

[1] E J Benveniste, Rigidity of isometric lattice actions on compact Riemannian manifolds, Geom. Funct. Anal. 10 (2000) 516

[2] L Burslem, A Wilkinson, Global rigidity of solvable group actions on $S^1$, Geom. Topol. 8 (2004) 877

[3] J Cantwell, L Conlon, An interesting class of $C^1$ foliations, Topology Appl. 126 (2002) 281

[4] D Damjanović, A Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\mathbb{Z}^k$ actions on the torus, Ann. of Math. 172 (2010) 1805

[5] D Fisher, First cohomology and local rigidity of group actions, to appear in Ann. of Math. (2)

[6] D Fisher, Local rigidity of group actions: past, present, future, from: "Dynamics, ergodic theory, and geometry" (editor B Hasselblatt), Math. Sci. Res. Inst. Publ. 54, Cambridge Univ. Press (2007) 45

[7] D Fisher, G Margulis, Almost isometric actions, property (T), and local rigidity, Invent. Math. 162 (2005) 19

[8] D Fisher, G Margulis, Local rigidity of affine actions of higher rank groups and lattices, Ann. of Math. 170 (2009) 67

[9] N Guelman, I Liousse, Actions of Baumslag–Solitar groups on surfaces

[10] N Guelman, I Liousse, $C^1$–actions of Baumslag–Solitar groups on $S^ 1$, Algebr. Geom. Topol. 11 (2011) 1701

[11] A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54, Cambridge Univ. Press (1995)

[12] A Katok, J Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math. 75 (1991) 203

[13] A Katok, R J Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova 216 (1997) 292

[14] A E Mccarthy, Rigidity of trivial actions of abelian-by-cyclic groups, Proc. Amer. Math. Soc. 138 (2010) 1395

[15] A Navas, Groupes résolubles de difféomorphismes de l'intervalle, du cercle et de la droite, Bull. Braz. Math. Soc. 35 (2004) 13

[16] V Niţică, A Török, Local rigidity of certain partially hyperbolic actions of product type, Ergodic Theory Dynam. Systems 21 (2001) 1213

[17] C Rivas, On spaces of Conradian group orderings, J. Group Theory 13 (2010) 337

[18] A Weil, Remarks on the cohomology of groups, Ann. of Math. 80 (1964) 149

[19] R J Zimmer, Lattices in semisimple groups and distal geometric structures, Invent. Math. 80 (1985) 123