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Journal of Lie Theory 31 (2021), No. 4, 1071--1084
Copyright Heldermann Verlag 2021

The Hilbert's Fifth Problem for Totally Intransitive Groupoids

Pawel Razny
Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

We continue the study of the Hilbert's fifth problem for groupoids by giving results concerning the totally intransitive case. We start by constructing a counterexample to the problem in its most general form. We then continue by noting the key feature of this example to give a positive answer to the problem under the additional assumptions that among the Lie algebras of the automorphism groups there is at most a finite collection of pairwise non-isomorphic Lie algebras and the base is of dimension 1. On the way we reduce the problem (for arbitrary dimension of the base) to smoothing a continuous Lie algebra bundle derived from the groupoid.

Keywords: Lie groupoids, topological groupoids.

MSC: 22A22.

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