Virtually free groups are stable in permutations (Q6062668)

In the paper under review, finitely generated groups which are stable in permutations (in short P-stable) are studied. It is proven that every finitely generated virtually free group is P-stable (see Theorem A). From this result, it follows that the modular group \(\mathrm{SL}_2(\mathbb{Z})\) is P-stable. It gives an answer to a question of A. Lubotzky. In the general case, P-stability is not preserved under direct products, for example, the group \(F_2 \times \mathbb{Z}\) is not P-stable. On the other side, the product groups \(F_2 \times (\mathbb{Z} / n \mathbb{Z})\) are P-stable for all \(n \in \mathbb{N}\), as follows from Theorem A. Stallings theorem on ends of groups implies that a finitely generated group \(G\) is virtually free if and only if \(G\) is isomorphic to the fundamental group \(\pi_1(\mathcal{G}, T)\) of a finite graph of groups \(\mathcal{G}\) with finite vertex groups with respect to some maximal spanning tree \(T\). Naturally associated to the graph of groups \(\mathcal{G}\) and the maximal spanning tree \(T\) there is another group \(\bar{\pi}_1(\mathcal{G}, T)\) admitting a quotient map \(\bar{\pi}_1(\mathcal{G}, T) \to \pi_1(\mathcal{G}, T)\). This group is isomorphic to the free product of the vertex groups of \(\mathcal{G}\) with the topological fundamental group of the underlying graph of \(\mathcal{G}\). As finite groups are P-stable, it follows immediately that the group \(\bar{\pi}_1(\mathcal{G}, T)\) is P-stable. Motivated by this, the authors introduce a relative notion of P-stable epimorphisms. In particular, a finitely generated group \(G\) is P-stable if and only if the natural epimorphism from the free group in the generators of \(G\) onto the group \(G\) is P-stable. Theorem A thereby reduces to the following statement, to which the major part of this work is dedicated: The epimorphism \(\bar{\pi}_1(\mathcal{G}, T) \to \pi_1(\mathcal{G}, T)\) is P-stable.