Simplicity of Neretin’s group of spheromorphisms. (English) Zbl 1050.20017

Kapoudjian, Christophe

Ann. Inst. Fourier 49, No. 4, 1225-1240 (1999).

Summary: Denote by \({\mathcal T}_n\), \(n\geq 2\), the regular tree whose vertices have valence \(n+1\), \(\partial {\mathcal T}_n\) its boundary. Yu. A. Neretin has proposed a group \(N_n\) of transformations of \(\partial {\mathcal T}_n\), 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 \(\text{Aut}({\mathcal T}_n)\) of tree automorphisms, and a Higman-Thompson group \(G_n\). We prove the simplicity of \(N_n\) and of a family of its subgroups.

MSC:

20E08	Groups acting on trees
20E32	Simple groups
22E65	Infinite-dimensional Lie groups and their Lie algebras: general properties
54H15	Transformation groups and semigroups (topological aspects)

Keywords:

Cantor set; Higman-Thompson groups; \(p\)-adic numbers; simple groups; spheromorphism; tree; tree automorphism group

References:

[1]	[1] , The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications, Kluwer Academic Publishers. · Zbl 0874.58005
[2]	[2] , 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. · Zbl 0613.20033
[3]	[3] , 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. · Zbl 0753.20007
[4]	[4] , and , Introductory notes on Richard Thompson’s groups, L’Enseignement Mathématique, 42 (1996), 215-256. · Zbl 0880.20027
[5]	[5] , The simplicity of certain groups of homeomorphisms, Compos. Math., 22 (1970), 165-173. · Zbl 0205.28201
[6]	[6] and , Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Mathematical Society Lecture Note Series 162, Cambridge University Press. · Zbl 1154.22301
[7]	[7] , 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]	[8] , Finitely presented infinite simple groups, Notes on pure mathematics, Australian National University, Canberra, 8 (1974). · Zbl 1479.20003
[9]	[3] , Untersuchungen über dire Grundlagen der Thermodynamik, Math. Ann., 67 (1909), 355-386.
[10]	[10] , Homological aspects and a Virasoro type extension for Higman-Thompson and Neretin groups almost-acting on trees, to appear. · Zbl 1064.20027
[11]	[11] and , 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. · Zbl 0286.02047
[12]	[12] , 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]	[13] , On combinatorial analogues of the group of diffeomorphisms of the circle, Russian Acad. Sci. Izv. Math., 41, n° 2 (1993). · Zbl 0789.22036
[14]	[3] , , Graph theory with applications, McMillan Press, 1976. · Zbl 0369.20013
[15]	[15] , 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. · Zbl 0214.51301