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Journal of Lie Theory 31 (2021), No. 1, 233--236
Copyright Heldermann Verlag 2021

On Compact Abelian Lie Groups of Homeomorphisms of Rm

Khadija Ben Rejeb
Higher Institute of Computer Science and Communication Technologies, University of Sousse, Hammam Sousse, Tunisia


Let $G$ be a compact Lie group of homeomorphisms of $\mathbb R^m$. The Naive conjecture saying that $G$ is conjugate to a subgroup of the orthogonal group $O(m)$ is known to be false for higher dimension. In this paper we give a partial answer by considering the action of the group $S = S(K_1) \times ... \times S(K_q)$ on $\mathbb R^m = K_1 \oplus ... \oplus K_q$, where $K_i = \mathbb R$ or $\mathbb C$ and $S(K_i) = \{x \!\in\! K_i : |x| = 1\}$ for $1\! \leq\! i \!\leq\! q$, and we show that $G$ is contained in $S$ if and only if every element of $G$ centralizes~$S$.

Keywords: Compact Lie group, homeomorphism of the Euclidean space Rm, conjugate, orthogonal group.

MSC: 37B05, 57S05, 57S10, 54H20, 37B20.

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