Opuscula Math. 38, no. 3 (2018), 395-408
https://doi.org/10.7494/OpMath.2018.38.3.395

Opuscula Mathematica

# On the boundedness of equivariant homeomorphism groups

Jacek Lech
Ilona Michalik
Tomasz Rybicki

Abstract. Given a principal $$G$$-bundle $$\pi:M\to B$$, let $$\mathcal{H}_G(M)$$ be the identity component of the group of $$G$$-equivariant homeomorphisms on $$M$$. The problem of the uniform perfectness and boundedness of $$\mathcal{H}_G(M)$$ is studied. It occurs that these properties depend on the structure of $$\mathcal{H}(B)$$, the identity component of the group of homeomorphisms of $$B$$, and of $$B$$ itself. Most of the obtained results still hold in the $$C^r$$ category.

Keywords: principal $$G$$-manifold, equivariant homeomorphism, uniformly perfect, bounded, $$C^r$$ equivariant diffeomorphism.

Mathematics Subject Classification: 57S05, 58D05, 55R91.

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