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

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 $\mathop{\mathrm{PSL}}(2,\mathbb{R})$.