Automorphism groups of rigid geometries on leaf spaces of foliations

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 $\mathcal{N}$.