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.

Full text (pdf)

1. K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. Ser. A 54 (1978), 53-54.
2. K. Abe, K. Fukui, On the structure of the group of Lipschitz homeomorphisms and its subgroups, J. Math. Soc. Japan 53 (2001), 501-511.
3. A. Banyaga, On the structure of the group of equivariant diffeomorphisms, Topology 16 (1977), 279-283.
4. A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and its Applications, vol. 400, Kluwer Academic Publishers Group, Dordrecht, 1997.
5. D. Burago, S. Ivanov, L. Polterovich, Conjugation invariant norms on groups of geometric origin, Adv. Stud. Pure Math. 52, Groups of Diffeomorphisms (2008), 221-250.
6. J.J. Duistermaat, J.A.C. Kolk, Lie groups, Springer-Verlag, 2000.
7. R.D. Edwards, R.C. Kirby, Deformations of spaces of imbeddings, Ann. Math. 93 (1971), 63-88.
8. D.B.A. Epstein, The simplicity of certain groups of homeomorphisms, Compositio Mathematica 22, Fasc. 2 (1970), 165-173.
9. M. Fraser, L. Polterovich, D. Rosen, On Sandon-type metrics for contactomorphism groups, D. Ann. Math. Québec (2017).
10. K. Fukui, On the uniform perfectness of equivariant diffeomorphism groups for principal $$G$$ manifolds, Opuscula Math. 37 (2017) 3, 381-388.
11. J.M. Gambaudo, E. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems 24 (1980) 5, 1591-1617.
12. A. Kowalik, T. Rybicki, On the homeomorphism groups of manifolds and their universal coverings, Cent. Eur. J. Math. 9 (2011) 6, 1217-1231.
13. A. Kriegl, P.W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, American Mathematical Society, 1997.
14. J.N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology 10 (1971), 297-298.
15. I. Michalik, T. Rybicki, On the structure of the commutator subgroup of certain homeomorphism groups, Topolology Appl. 158 (2011), 1314-1324.
16. T. Rybicki, On commutators of equivariant homeomorphisms, Topology Appl. 154 (2007), 1561-1564.
17. T. Rybicki, Boundedness of certain automorphism groups of an open manifold, Geometriae Dedicata 151 (2011) 1, 175-186.
18. L.C. Siebenmann, Deformation of homeomorphisms on stratified sets, I, II, Comment. Math. Helv. 47 (1972), 123-163.
19. T. Tsuboi, On the uniform perfectness of diffeomorphism groups, Advanced Studies in Pures Math. 52, Groups of Diffeomorphisms (2008), 505-524.
• Jacek Lech
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
• Ilona Michalik
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
• Tomasz Rybicki
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
• Communicated by P.A. Cojuhari.
• Revised: 2018-01-23.
• Accepted: 2018-02-02.
• Published online: 2018-03-19.