Compact presentability of tree almost automorphism groups

Le Boudec, Adrien

Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 329-365.

Nous prouvons que certains groupes de presqu’automorphismes d’arbres sont compactement présentés. Parmi ces groupes figurent le groupe de Neretin des sphéromorphismes d’un arbre régulier, ainsi que le groupe topologiquement simple contenant le complété profini du groupe de Grigorchuk construit par Barnea, Ershov et Weigel.

Nous montrons de plus que la fonction de Dehn de ces groupes est asymptotiquement bornée par la fonction de Dehn du groupe de Higman–Thompson. Combiné à un résultat de Guba, cela implique que la fonction de Dehn du groupe de Neretin de l’arbre trivalent est polynomialement bornée.

We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin’s group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel.

We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman–Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.

Classification : 20E08, 20F65, 20E32
Keywords: Almost automorphisms of trees, Neretin group, compact presentability, Dehn function
Mot clés : Presqu’automorphismes d’arbres, groupe de Neretin, présentation compacte, fonction de Dehn

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