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Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle. (English) Zbl 0852.57031
The author defines certain groups of Lipschitz homeomorphism on the circle (and on $$\mathbb{R}$$ with compact support) and studies the relation between dynamics, in the form of Pixton actions inside the homeomorphism groups, and homology of the classifying spaces associated to the homeomorphism groups. In particular, the author shows that there is a Pixton action inside certain of these groups (depending on size and dimension) and that, for the $$\mathbb{R}$$ case, classifying spaces become more and more acyclic as a certain parameter $$\beta$$ increases. It is also shown, in the circle case, that the first homology of the homeomorphism groups vanishes.

##### MSC:
 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
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