Automorphism groups of rigid geometries on leaf spaces of foliations

Abstract

We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category ${\mathfrak{F}}_0$ of leaf manifolds containing the orbifold category as a full subcategory. Objects of ${\mathfrak{F}}_0$ may have non-Hausdorff topology unlike the orbifolds. The topology of some objects of ${\mathfrak{F}}_0$ does not satisfy the separation axiom $T_0$. It is shown that for every $\mathcal{N}\in Ob({\mathfrak{F}}_0)$ a rigid geometry $\zeta$ on $\mathcal{N}$ admits a desingularization. Moreover, for every such $\mathcal{N}$ we prove the existence and the uniqueness of a finite dimensional Lie group structure on the automorphism group $Aut(\zeta )$ of the rigid geometry $\zeta$ on $\mathcal{N}$.