The uniqueness of the actions of certain branch groups on rooted trees.
The authors consider the case of groups `of branch type' which act on rooted trees. A rooted tree of type \((l_n)\) is a tree \(T\) with a vertex \(v_0\) of valency \(l_0\), such that every vertex at distance \(n\geq 1\) from \(v_0\) has valency \(l_n+1\), where \((l_n)\) is a sequence of integers with \(l_n\geq 2\) for each \(n\). The authors consider a branch group \(G\) which acts on the tree \(T\) and satisfies the additional conditions: a) for each vertex \(u\) of \(T\) the stabilizer of \(u\) in \(G\) acts as a (transitive) cyclic group of prime order on the edges descending from \(u\); b) whenever \(u,u'\) are incomparable vertices and \(v<u\), there is an element \(g\in G\) with \(u'g=u'\) and \(vg\neq v\). It is proved (Theorem 1) that if \(T'\) is another rooted tree on which \(G\) acts as a branch group then there is an isomorphism preserving the action of \(G\) from \(T'\) to a tree obtained from \(T\) by deletion of layers. Also other results are obtained about these branch groups. Further, the authors verify that condition b) holds for most of the well-known examples of branch groups.
scientific article; zbMATH DE number 2002026