Free subgroups in the automorphism groups of trees. (Sous-groupes libres dans les groupes d’automorphismes d’arbres.) (French) Zbl 0744.20024

Pays, Isabelle; Valette, Alain

Enseign. Math., II. Sér. 37, No. 1-2, 151-174 (1991).

Let \(G\) be a group acting (effectively) on a (locally finite) tree \(X\). The main purpose of this paper is to give a concise and well readable proof of the equivalence of the following properties of \(G\): (i) \(G\) has no fixed vertex, edge or end in \(X\). (ii) \(G\) contains a free nonabelian subgroup acting freely on \(X\). (iii) the closure \(\bar G\) of \(G\) in \(\hbox{Aut }X\) is not amenable. All implications in this theorem were known previously, although scattered in several papers [see in particular C. Nebbia: Pac. J. Math. 135, No. 2, 371-380 (1988; Zbl 0671.43003) and H. Culler, J. W. Morgan: Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)], but part of the proofs are new. The relation of the theorem with the Tits alternative is also explained.

Reviewer: F.Herrlich (Karlsruhe)

MSC:

20E08	Groups acting on trees
20B27	Infinite automorphism groups
20E07	Subgroup theorems; subgroup growth

Keywords:

group actions; automorphisms of trees; amenability; locally finite tree; free nonabelian subgroup; Tits alternative

Citations:

Zbl 0671.43003; Zbl 0658.20021