Actions of some infinite Lie groups (Q1105240)

Let \(\Gamma\) be the inverse limit of a system \((G_ r)_{r=1,2,...}\) of finite dimensional Lie groups and \(H_ r\) the kernel of the canonical projection \(\Gamma \to G_ r\). Assume that \(\Gamma\) acts continuously on a finite dimensional metric space M and \(H_ r\) is connected for each r. Let \(x\in M\). Under the above assumptions, \textit{D. B. A. Epstein} and \textit{W. Thurston} [Proc. Lond. Math. Soc., III. Ser. 38, 219-237 (1979; Zbl 0409.58001)] have proved that for sufficiently large r, \(H_ r\) stabilizes x. The author proves a similar theorem for a wider class of such groups, relaxing the connectedness hypothesis about \(H_ r\). In order to do this, he defines the concept of infinite Lie group of type WL. The metric space M is replaced by a smooth finite dimensional manifold, not necessarily metrizable. Using this modified version of the theorem, he shows that natural functors over the category of n- dimensional analytic manifolds with analytic embeddings as morphisms have finite order.