2021, 17: 111-143. doi: 10.3934/jmd.2021004

Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan

Received  May 22, 2019 Revised  November 15, 2020 Published  March 2021

Let $ G $ be the group of orientation-preserving isometries of a rank-one symmetric space $ X $ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $ \Gamma \subset G $ on the boundary of $ X $, which is diffeomorphic to a sphere. When $ X $ is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.

Citation: Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004
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show all references

References:
[1]

M. Asaoka, Rigidity of certain solvable actions on the sphere, Geom. Topol., 16 (2012), 1835-1857.  doi: 10.2140/gt.2012.16.1835.  Google Scholar

[2]

M. Asaoka, Rigidity of certain solvable actions on the torus, Geometry, Dynamics, and Foliations, (2013), 269–281. doi: 10.2969/aspm/07210269.  Google Scholar

[3]

L. Burslem and A. Wilkinson, Global rigidity of solvable group actions on $S^1$, Geom. Topol., 8 (2004), 877-924.  doi: 10.2140/gt.2004.8.877.  Google Scholar

[4]

W. Casselman and M. S. Osborne, The $n$-cohomology of the representations with an infinitesimal character, Compositio Math., 31 (1975), 219-227.   Google Scholar

[5]

D. Fisher, Recent progress in the Zimmer program, preprint, arXiv: 1711.07089. Google Scholar

[6]

É. Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 163-185.   Google Scholar

[7]

A. W. Knapp, Lie Groups Beyond an Introduction, 2$^{nd}$ edition, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4757-2453-0.  Google Scholar

[8]

M. Okada, Local rigidity of certain actions of nilpotent-by-cyclic groups on the sphere, J. Math. Sci. Univ. Tokyo, 26 (2019), 15-53.   Google Scholar

[9]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[10]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.  doi: 10.2307/2372437.  Google Scholar

[11]

D. Stowe, The stationary set of a group action, Proc. Amer. Math. Soc., 79 (1980), 139-146.  doi: 10.1090/S0002-9939-1980-0560600-2.  Google Scholar

[12]

D. A. Vogan, Representations of Real Reductive Lie Groups, Progress in Mathematics, 15, Birkhäuser, Boston, Mass., 1981.  Google Scholar

[13]

A. Wilkinson and J. Xue, Rigidity of some abelian-by-cyclic solvable group actions on $\mathbb{T}^N$, Comm. Math. Phys., 376 (2020), 1223-1259.  doi: 10.1007/s00220-019-03658-3.  Google Scholar

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