Simplicity of Neretin's group of spheromorphisms

Christophe Kapoudjian

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 4, page 1225-1240
  • ISSN: 0373-0956

Abstract

top
Denote by 𝒯 n , n ≥ 2 , the regular tree whose vertices have valence n + 1 , ∂ 𝒯 n its boundary. Yu. A. Neretin has proposed a group N n of transformations of ∂ 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.

How to cite

top

Kapoudjian, Christophe. "Simplicity of Neretin's group of spheromorphisms." Annales de l'institut Fourier 49.4 (1999): 1225-1240. <http://eudml.org/doc/75379>.

@article{Kapoudjian1999,
abstract = {Denote by $\{\cal T\}_n$, $n\ge 2$, the regular tree whose vertices have valence $n+1$, $\partial \{\cal T\}_n$ its boundary. Yu. A. Neretin has proposed a group $N_n$ of transformations of $\partial \{\cal T\}_n$, 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 $\{\rm Aut\}(\{\cal T\}_n)$ of tree automorphisms, and a Higman-Thompson group $G_n$. We prove the simplicity of $N_n$ and of a family of its subgroups.},
author = {Kapoudjian, Christophe},
journal = {Annales de l'institut Fourier},
keywords = {Cantor set; Higman-Thompson groups; -adic numbers; simple groups; spheromorphism; tree; tree automorphism group},
language = {eng},
number = {4},
pages = {1225-1240},
publisher = {Association des Annales de l'Institut Fourier},
title = {Simplicity of Neretin's group of spheromorphisms},
url = {http://eudml.org/doc/75379},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Kapoudjian, Christophe
TI - Simplicity of Neretin's group of spheromorphisms
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 4
SP - 1225
EP - 1240
AB - Denote by ${\cal T}_n$, $n\ge 2$, the regular tree whose vertices have valence $n+1$, $\partial {\cal T}_n$ its boundary. Yu. A. Neretin has proposed a group $N_n$ of transformations of $\partial {\cal T}_n$, 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 ${\rm Aut}({\cal T}_n)$ of tree automorphisms, and a Higman-Thompson group $G_n$. We prove the simplicity of $N_n$ and of a family of its subgroups.
LA - eng
KW - Cantor set; Higman-Thompson groups; -adic numbers; simple groups; spheromorphism; tree; tree automorphism group
UR - http://eudml.org/doc/75379
ER -

References

top
  1. [1] A. BANYAGA, The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications, Kluwer Academic Publishers. Zbl0874.58005
  2. [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. Zbl0613.20033MR88m:20110
  3. [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. Zbl0753.20007MR94f:20059
  4. [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. Zbl0880.20027MR98g:20058
  5. [5] D.B.A. EPSTEIN, The simplicity of certain groups of homeomorphisms, Compos. Math., 22 (1970), 165-173. Zbl0205.28201MR42 #2491
  6. [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. Zbl1154.22301
  7. [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). Zbl0217.49602
  8. [8] G. HIGMAN, Finitely presented infinite simple groups, Notes on pure mathematics, Australian National University, Canberra, 8 (1974). 
  9. [9] C. KAPOUDJIAN, Sur des généralisations p-adiques du groupe des difféomorphismes du cercle, Thèse de Doctorat, Université Claude Bernard Lyon1, décembre 1998. 
  10. [10] C. KAPOUDJIAN, Homological aspects and a Virasoro type extension for Higman-Thompson and Neretin groups almost-acting on trees, to appear. 
  11. [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. Zbl0286.02047MR53 #629
  12. [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. Zbl0576.22007
  13. [13] Yu.A. NERETIN, On combinatorial analogues of the group of diffeomorphisms of the circle, Russian Acad. Sci. Izv. Math., 41, n° 2 (1993). Zbl0789.22036
  14. [14] J.-P. SERRE, Arbres, amalgames, SL2, Astérisque 46. Zbl0369.20013MR57 #16426
  15. [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. Zbl0214.51301MR45 #8582

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.