# Automorphism groups of rigid geometries on leaf spaces of foliations

## TL;DR

This paper develops a framework for rigid geometries on leaf spaces of foliations, including non-Hausdorff cases, and proves the existence and uniqueness of Lie group structures on their automorphism groups.

## Contribution

It introduces a new category of leaf space geometries, extends the automorphism group theory to non-Hausdorff cases, and establishes a Lie group structure on automorphisms.

## Key findings

- Automorphism groups of these geometries admit finite-dimensional Lie group structures.
- Existence and uniqueness of desingularizations for geometries on leaf spaces.
- Extension of orbifold theory to more general leaf space geometries.

## 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 $\mathcal{N}$.