Let $G$ be a finite subgroup of the group $H$ of orientation-preserving homeomorphisms of the plane that fix the origin. Is $G$ conjugate in $H$ to a group of rotations?

I've been told this result was proved in the 1930s, perhaps by Helmut Kneser, and also that it is due to Kerekjarto, Brouwer and Eilenberg, probably not in the same paper.

If $G$ consists of diffeomorphisms, this is a special case of a theorem of Bochner. Can someone supply a reference, or a counter-example?


1 Answer 1


See, say, here for the case of finite cyclic groups of (orientation preserving) homeomorphisms of the plane. It follows from this result (due to Kerekjarto et al) that the fixed-point set in $R^2$ of such a cyclic group is a single point. For an arbitrary finite group $G< Homeo_+(R^2)$, we conclude that $S^2/G$ is a compact oriented topological orbifold $O$ with finite fundamental group. (Here $S^2$ is the 1-point compactification of $R^2$.) Now apply the classification of 2d orbifolds to conclude that $O$ is homeomorphic to the quotient of $S^2$ by a finite group of rotations. Since the point $\infty$ is fixed by $G$, conclude that $O$ has finite cyclic fundamental group, this $G$ is cyclic.


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