Simplicity of Neretin's group of spheromorphisms

Kapoudjian, Christophe

Annales de l'Institut Fourier, Tome 49 (1999), p. 1225-1240 / Harvested from Numdam

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Abstract

Notons $𝒯_{n}$ , $n \geq 2$ , l’arbre régulier dont les sommets sont de valence $n + 1$ , $\partial 𝒯_{n}$ son bord. Yu. A. Neretin a proposé, comme analogue combinatoire du groupe des difféomorphismes du cercle, un groupe de transformations $N_{n}$ agissant sur $\partial 𝒯_{n}$ . On montre que $N_{n}$ est engendré par deux groupes: le groupe $Aut (𝒯_{n})$ des automorphismes de l’arbre, et un groupe de Higman-Thompson $G_{n}$ . On prouve la simplicité de $N_{n}$ et d’une famille de ses sous-groupes.

Denote by $𝒯_{n}$ , $n \geq 2$ , the regular tree whose vertices have valence $n + 1$ , $\partial 𝒯_{n}$ its boundary. Yu. A. Neretin has proposed a group $N_{n}$ of transformations of $\partial 𝒯_{n}$ , thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that $N_{n}$ is generated by two groups: the group $Aut (𝒯_{n})$ of tree automorphisms, and a Higman-Thompson group $G_{n}$ . We prove the simplicity of $N_{n}$ and of a family of its subgroups.

Publié le : 1999-01-01

MR 2001b:20070 | Zbl 01323235

DOI : https://doi.org/10.5802/aif.1715

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     author = {Kapoudjian, Christophe},
     title = {Simplicity of Neretin's group of spheromorphisms},
     journal = {Annales de l'Institut Fourier},
     volume = {49},
     year = {1999},
     pages = {1225-1240},
     doi = {10.5802/aif.1715},
     mrnumber = {2001b:20070},
     zbl = {01323235},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1999__49_4_1225_0}
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Kapoudjian, Christophe. Simplicity of Neretin's group of spheromorphisms. Annales de l'Institut Fourier, Tome 49 (1999) pp. 1225-1240. doi : 10.5802/aif.1715. http://gdmltest.u-ga.fr/item/AIF_1999__49_4_1225_0/

Bibliographie

[1] A. Banyaga, The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications, Kluwer Academic Publishers. | Zbl 0874.58005

[2] K.S. Brown, Finiteness properties of groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, I. 11, 1985) J. Pure Appl. Algebra 1-3 (1987), 45-75. | MR 88m:20110 | Zbl 0613.20033

[3] K.S. Brown, The Geometry of Finitely Presented Infinite Simple Groups, Algorithms and classifications in combinatorial group theory (Berkeley, A, 1989), Math. Sci. Res. Inst. Publ., 23, Springer, New-York (1992), 121-136. | MR 94f:20059 | Zbl 0753.20007

[4] J.W. Cannon, W.J. Floyd and W.R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathématique, 42 (1996), 215-256. | MR 98g:20058 | Zbl 0880.20027

[5] D.B.A. Epstein, The simplicity of certain groups of homeomorphisms, Compos. Math., 22 (1970), 165-173. | Numdam | MR 42 #2491 | Zbl 0205.28201

[6] A. Figà-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Mathematical Society Lecture Note Series 162, Cambridge University Press.

[7] M.-R. Herman, Simplicité du groupe des difféomorphismes de classe C∞, isotopes à l'identité, du tore de dimension n, C. R. Acad. Sc. Paris, 273 (26 juillet 1971). | Zbl 0217.49602

[8] G. Higman, Finitely presented infinite simple groups, Notes on pure mathematics, Australian National University, Canberra, 8 (1974).

[9] C. Kapoudjian, Sur des généralisations p-adiques du groupe des difféomorphismes du cercle, Thèse de Doctorat, Université Claude Bernard Lyon1, décembre 1998.

[10] C. Kapoudjian, Homological aspects and a Virasoro type extension for Higman-Thompson and Neretin groups almost-acting on trees, to appear.

[11] R. Mckenzie and R.J. Thompson, An elementary construction of unsolvable word problems in group theory, in “Word problems”, Proc. Conf. Irvine 1969 (edited by W.W. Bone, F.B. Cannonito, and R.C. Lyndon), Studies in Logic and the Foundations of Mathematics, 71 (1973), North-Holland, Amsterdam, 457-478. | MR 53 #629 | Zbl 0286.02047

[12] Yu.A. Neretin, Unitary representations of the diffeomorphism group of the p-adic projective line, translated from Funktsional'nyi Analiz i Ego Prilozheniya, 18, n° 4 (1984), 92-93. | Zbl 0576.22007

[13] Yu.A. Neretin, On combinatorial analogues of the group of diffeomorphisms of the circle, Russian Acad. Sci. Izv. Math., 41, n° 2 (1993). | Zbl 0789.22036

[14] J.-P. Serre, Arbres, amalgames, SL2, Astérisque 46. | MR 57 #16426 | Zbl 0369.20013

[15] J. Tits, Sur le groupe des automorphismes d'un arbre, Essays on topology and related topics, Mémoires dédiés à G. de Rham, Springer-Verlag, Berlin (1970), 188-211. | MR 45 #8582 | Zbl 0214.51301