## Sur un groupe remarquable de difféomorphismes du cercle. (On a remarkable group of the diffeomorphisms of the circle).(French)Zbl 0647.58009

Let $$\tilde G$$ be the group of homeomorphisms g of $${\mathbb{R}}$$ such that: 1. $$g(x+1)=g(x)+1$$; 2. g is piecewise affine; 3. if x is a point of discontinuity of the derivative of g, then x and g(x) are of the form $$p\cdot 2^ q$$ with $$p,q\in {\mathbb{Z}}$$; 4. the one-sided derivatives of g at every point are of the form $$2^ q$$ with $$q\in {\mathbb{Z}}$$; 5. g(0) is of the form $$p\cdot 2^ q$$ with $$p,q\in {\mathbb{Z}}$$. Due to the property 1 every element of $$\tilde G$$ defines a homeomorphism of $$S^ 1={\mathbb{R}}/{\mathbb{Z}}$$. Let G be the group of these homeomorphisms of $$S^ 1$$. This group G is the subject of the paper.
The first result asserts that there exist a homeomorphism h of $$S^ 1$$ such that $$hGh^{-1}$$ is a group of $$C^{\infty}$$ diffeomorphisms. This justifies the title. The paper is divided into two parts, devoted, respectively, to cohomological properties of G and to qualitative properties of representations $$G\to \text{Homeo}(S^ 1)$$. In both parts many interesting results are proved. I describe only some of them.
Theorem. There exist homology equivalencies $$B\tilde G\to (S^ 3)^{S^ 1}$$, $$BG\to (S^ 3)^{S^ 1}\times_{S^ 1}ES^ 1$$. (Here BH denotes the classifying space of H, $$X^{S^ 1}$$ denotes the space of free loops in X, and $$ES^ 1$$ is the total space of the universal $$S^ 1$$-bundle.) This theorem allows tocompute the cohomology of $$\tilde G,G$$. The generators of the cohomology ring are interpreted as an Euler class and an analogue of the Godbillon-Ven class.
It is proved also that G is isomorphic to a discrete subgroup of the group of $$C^ 1$$ diffeomorphisms of $$S^ 1$$ (with $$C^ 1$$ topology). Another interesting result asserts that every nontrivial representation $$G\to Diff^ 2(S^ 1)$$ (diffeomorphisms of the class $$C^ 2$$) is semiconjugate to the inclusion $$G\text{Homeo}(S^ 1)$$.
Finally, it is necessary to note that the group G was already considered in several parts of mathematics: logic, algebra, homotopy theory. The authors argue that their results support an analogy between G and arithmetic groups. This is a truly remarkable group.
Reviewer: N.V.Ivanov

### MSC:

 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 20F38 Other groups related to topology or analysis

### Keywords:

diffeomorphisms of the circle; cohomology; foliations
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