Simplicity of Neretin's group of spheromorphisms

Kapoudjian, Christophe

Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1225-1240.

Notons ${𝒯}_n$, $n\ge 2$, 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 $\mathrm{Aut}\left({𝒯}_n\right)$ 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 ${𝒯}_n$, $n\ge 2$, 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 $\mathrm{Aut}\left({𝒯}_n\right)$ of tree automorphisms, and a Higman-Thompson group $G_n$. We prove the simplicity of $N_n$ and of a family of its subgroups.

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