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Local rigidity of weak or no hyperbolicity algebraic actions

Zhenqi Jenny Wang

Abstract

In this paper we study rigidity properties of abelian \hyphenation{break-able}actions with weak or no hyperbolicty. We introduce a general strategy for proving $C^\infty$ local rigidity of algebraic actions. As a consequence, we show $C^\infty$ local rigidity for a broad class of parabolic algebraic actions on homogeneous spaces of semisimple Lie groups. This is the first time in the literature that (strong) local rigidity for these actions is addressed.

Local rigidity of weak or no hyperbolicity algebraic actions

Abstract

In this paper we study rigidity properties of abelian \hyphenation{break-able}actions with weak or no hyperbolicty. We introduce a general strategy for proving local rigidity of algebraic actions. As a consequence, we show local rigidity for a broad class of parabolic algebraic actions on homogeneous spaces of semisimple Lie groups. This is the first time in the literature that (strong) local rigidity for these actions is addressed.

Paper Structure

This paper contains 74 sections, 44 theorems, 432 equations.

Table of Contents

  1. Introduction
  2. Abelian algebraic actions
  3. Rigidity of actions and related notions
  4. History and motivation
  5. Results of the paper
  6. Proof strategy
  7. Construction of inverses of coboundary operators
  8. Splitting for $u$
  9. Higher rank trick
  10. Directional smoothing operator
  11. Construction of $\Theta$, part I
  12. Construction of $\Theta$, part II
  13. Convergence in the iteration
  14. Scope of the strategy
  15. Notations and preparatory steps
  16. ...and 59 more sections

Key Result

Theorem 1.1

Suppose $\mathbb{G}\neq \mathbb{G}_1$. Let $A\subseteq G$ be a closed abelian subgroup of $\mathbb{G}$ with the following property: Then there is $\ell\in \mathbb{N}$ such that the action $\alpha_A$ is $C^{\infty,\ell,\infty}$locally rigid.

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Example 1
  • Example 2
  • Proposition 2.1
  • ...and 70 more