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**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.

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