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The orbit-fixing deformation spaces of an action of a Lie groupoid

Hirokazu Maruhashi

Abstract

The orbit-fixing deformation spaces of $C^\infty$ locally free actions of simply connected Lie groups on closed $C^\infty$ manifolds have been studied by several authors. In this paper we reformulate the deformation space by imitating the Teichmüller space of a surface. The new formulation seems to be more appropriate for actions of Lie groups which are not simply connected. We also consider actions which may not be locally free, and generalize the deformation spaces for actions of Lie groupoids. Furthermore by using bornologies on Lie groupoids, we make the definition of the deformation space more suitable to deal with actions on noncompact manifolds. In this generality we prove that "cocycle rigidity" implies the deformation space is a point. We compute the deformation space of the action of $\mathop{\mathrm{PSL}}(2,\mathbb{R})$ on $Γ\backslash\mathop{\mathrm{PSL}}(2,\mathbb{R})$ by right multiplication for a torsion free cocompact lattice $Γ$ in $\mathop{\mathrm{PSL}}(2,\mathbb{R})$.

The orbit-fixing deformation spaces of an action of a Lie groupoid

Abstract

The orbit-fixing deformation spaces of locally free actions of simply connected Lie groups on closed manifolds have been studied by several authors. In this paper we reformulate the deformation space by imitating the Teichmüller space of a surface. The new formulation seems to be more appropriate for actions of Lie groups which are not simply connected. We also consider actions which may not be locally free, and generalize the deformation spaces for actions of Lie groupoids. Furthermore by using bornologies on Lie groupoids, we make the definition of the deformation space more suitable to deal with actions on noncompact manifolds. In this generality we prove that "cocycle rigidity" implies the deformation space is a point. We compute the deformation space of the action of on by right multiplication for a torsion free cocompact lattice in .

Paper Structure

This paper contains 82 sections, 165 theorems, 381 equations.

Table of Contents

  1. Introduction
  2. Two different definitions of parameter rigidity
  3. The definition in MM
  4. The definition in Ma
  5. Lie groupoids
  6. Lie groupoids
  7. Morphisms of Lie groupoids
  8. The identity component of a Lie groupoid
  9. Actions of Lie groupoids
  10. Actions of Lie groupoids
  11. Action morphisms and action groupoids
  12. (Semi)conjugacies and (constant) cocycles
  13. Action of an action groupoid
  14. Action groupoid morphisms
  15. The principal bundle structures on the target fibers
  16. ...and 67 more sections

Key Result

Proposition A

Let ${\mathcal{G}}\rightrightarrows M$ be an $s$-simply connected Lie groupoid, $N_0$ be a $C^\infty$ manifold, $\nu_0\colon N_0\to M$ be a surjective submersion and $\rho_0$ be a $C^\infty$ locally free right action of ${\mathcal{G}}$ on $\nu_0$. Then we have

Theorems & Definitions (548)

  • Proposition A
  • Proposition B
  • Theorem C
  • Theorem D: Semiconjugacy-to-Conjugacy Theorem for $bnt$
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Lemma 6
  • ...and 538 more