- Proper Lie groupoids and their orbit spaces
- Award date
- 25 September 2018
- Number of pages
- Document type
- PhD thesis
- Faculty of Science (FNWI)
- Korteweg-de Vries Institute for Mathematics (KdVI)
This thesis studies proper Lie groupoids on three levels: the groupoids themselves, their induced foliations, and their orbit spaces. Proper Lie groupoids are shown to admit desingularisations via a successive blow-up procedure, whereby orbits are systematically added to achieve regularity. Regarding their foliations, a thorough treatment of the known integration results for singular foliations is included. Moreover, the underlying orbit spaces of proper Lie groupoids are studied. This is done by first providing an intrinsic definition of a so-called orbispace using atlases, both in the language of Morita bibundles, and in the language of Morita equivalences through fractions. An orbispace is said to be proper if it admits a proper defining atlas. It is then shown that proper Lie groupoids, up to a precise notion of Morita equivalence, correspond exactly to such proper orbispaces. This can be interpreted as the statement that proper orbispaces form a subcategory of all differentiable stacks. All of these developments mirror the well-known correspondences between regular proper Lie groupoids, regular foliations, and orbifolds. The above results are further shown to hold in the setting of proper Riemannian groupoids. In particular the desingularisation procedure can be performed in such a way that the regularised groupoid has arbitrarily small Gromov—Hausdorff distance from the original groupoid. Moreover, proper Riemannian orbispaces are defined and shown to correspond precisely to appropriate equivalence classes of proper Riemannian groupoids. This thesis contains various other results, including those on holonomy groupoids of orbit-like foliations, and a de Rham theorem for orbispaces.
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