Free products of finite groups and groups of finitely automatic permutations (Q2783058)

Let \(X\) be a finite alphabet. The free monoid on \(X\) is denoted by \(X^*\) and the set of all infinite sequences on \(X\) is denoted by \(X^\omega\). The set \(X^*\) can be considered a rooted tree \(T\) where \(uv\) is an edge iff there is \(x\in X\) such that \(u=vx\) or \(v=ux\) and the root is the empty word. The set of end points of \(T\) can be identified with the set \(X^\omega\). The automorphism group \(\Aut(T)\) of the rooted tree \(T\) is isomorphic to the wreath product of an infinite sequence of symmetric groups \(S_n\) where \(n\) is the cardinality of \(X\). Every automorphism \(f\in\Aut(T)\) determines a permutation \(\pi_f\) on \(X^\omega\), the boundary of \(T\). Such permutations of \(X^\omega\) are called automatic. The author gives a constructive embedding of the free product \(G_1*G_2*\cdots*G_k\) of finite groups into the group of automatic permutations of \(X^\omega\) where the cardinality of \(X\) is the same as the order of the largest of the finite groups \(G_1,G_2,\dots,G_k\).NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].