On closed subgroups of the group of homeomorphisms of a manifold
Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 147-159.

Let $M$ be a triangulable compact manifold. We prove that, among closed subgroups of ${\mathrm{Homeo}}_{0}\left(M\right)$ (the identity component of the group of homeomorphisms of $M$), the subgroup consisting of volume preserving elements is maximal.

Soit $M$ une variété triangulable compacte. Nous montrons que, parmi les sous-groupes de ${\mathrm{Homeo}}_{0}\left(M\right)$ (composante connexe de l’identité du groupe des homéomorphismes de $M$), le sous-groupe des homéomorphismes préservant le volume est maximal.

DOI: 10.5802/jep.7
Classification: 57S05,  57M60,  37E30
Keywords: Transformation groups, homeomorphisms, maximal closed subgroups
Le Roux, Frédéric 1

1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Pierre et Marie Curie 4, place Jussieu, Case 247, 75252 Paris Cedex 5, France
@article{JEP_2014__1__147_0,
author = {Le Roux, Fr\'ed\'eric},
title = {On closed subgroups of the group of homeomorphisms of a manifold},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {147--159},
publisher = {Ecole polytechnique},
volume = {1},
year = {2014},
doi = {10.5802/jep.7},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jep.7/}
}
TY  - JOUR
AU  - Le Roux, Frédéric
TI  - On closed subgroups of the group of homeomorphisms of a manifold
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2014
SP  - 147
EP  - 159
VL  - 1
PB  - Ecole polytechnique
UR  - http://www.numdam.org/articles/10.5802/jep.7/
UR  - https://doi.org/10.5802/jep.7
DO  - 10.5802/jep.7
LA  - en
ID  - JEP_2014__1__147_0
ER  - 
%0 Journal Article
%A Le Roux, Frédéric
%T On closed subgroups of the group of homeomorphisms of a manifold
%J Journal de l’École polytechnique — Mathématiques
%D 2014
%P 147-159
%V 1
%I Ecole polytechnique
%U https://doi.org/10.5802/jep.7
%R 10.5802/jep.7
%G en
%F JEP_2014__1__147_0
Le Roux, Frédéric. On closed subgroups of the group of homeomorphisms of a manifold. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 147-159. doi : 10.5802/jep.7. http://www.numdam.org/articles/10.5802/jep.7/

[Bes04] Bestvina, M. Questions in geometric group theory, collected by M. Bestvina (2004) (http://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)

[Bro62] Brown, M. A mapping theorem for untriangulated manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 92-94 | MR | Zbl

[Fat80] Fathi, A. Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 1, pp. 45-93 | Numdam | MR | Zbl

[Ghy01] Ghys, É. Groups acting on the circle, Enseign. Math. (2), Volume 47 (2001) no. 3-4, pp. 329-407 | MR | Zbl

[GM06] Giblin, J.; Markovic, V. Classification of continuously transitive circle groups, Geom. Topol., Volume 10 (2006), pp. 1319-1346 | DOI | MR | Zbl

[GP75] Goffman, C.; Pedrick, G. A proof of the homeomorphism of Lebesgue-Stieltjes measure with Lebesgue measure, Proc. Amer. Math. Soc., Volume 52 (1975), pp. 196-198 | MR | Zbl

[Kir69] Kirby, R. C. Stable homeomorphisms and the annulus conjecture, Ann. of Math. (2), Volume 89 (1969), pp. 575-582 | MR | Zbl

[KT13] Kwakkel, F.; Tal, F. Homogeneous transformation groups of the sphere (2013) (arXiv:1309.0179v1)

[Nav07] Navas, A. Grupos de difeomorfismos del círculo, Ensaios Matemáticos, 13, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007, pp. ii+249 | Zbl

[OU41] Oxtoby, J. C.; Ulam, S. M. Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), Volume 42 (1941), pp. 874-920 | MR | Zbl

[Qui82] Quinn, F. Ends of maps. III. Dimensions $4$ and $5$, J. Differential Geom., Volume 17 (1982) no. 3, pp. 503-521 http://projecteuclid.org/euclid.jdg/1214437139 | MR | Zbl

Cited by Sources: