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

TL;DR

This paper generalizes the concept of orbit-fixing deformation spaces from locally free actions of simply connected Lie groups to more complex actions of Lie groupoids, using Teichmüller space analogies and bornologies, and computes specific examples.

Contribution

It introduces a new formulation of deformation spaces for Lie group actions, extending to non-simply connected groups, non-locally free actions, and noncompact manifolds, with applications to specific group actions.

Findings

Deformation space is a point under cocycle rigidity.

Reformulation analogous to Teichmüller space improves understanding.

Explicit computation for PSL(2,R) action on quotient space.

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 Teichm\"{u}ller 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 $\Gamma\backslash\mathop{\mathrm{PSL}}(2,\mathbb{R})$ by right multiplication for a torsion free cocompact lattice $\Gamma$ in $\mathop{\mathrm{PSL}}(2,\mathbb{R})$.