Strict Dead End Elements in Free Soluble Groups

Let G be a group generated by a finite set A. An element g ∈ G is a strict dead end of depth k (with respect to A) if |g| > |ga1| > |ga1a2| > · · · > |ga1a2 · · · ak| for any a1, a2, . . . , ak ∈ A such that the word a1a2 · · ·ak is freely irreducible. (Here |g| is the distance from g to the identity in the Cayley graph of G.) We show that in finitely generated free soluble groups of degree d ≥ 2 there exist strict dead elements of depth k = k(d), which grows exponentially with respect to d. Let G be a group generated by a finite set A. By C = C(G,A) we denote the right Cayley graph of G with respect to A. The set of vertices of this graph is G, the set of edges is G × A. Every edge e = (g, a), where g ∈ G, a ∈ A, starts in g and ends in ga. This edge is labelled by a. An inverse edge is e = (ga, a). For every g ∈ G let |g| denote the distance in C from the vertex g to the identity. An element g is a dead end in C whenever |g| > |ga| for all a ∈ A. This concept was introduced by Bogopolski in [3]. One of the easiest examples is the following. Let us consider the group G = Z×Z2, where a, b generate the factors. It is easy to see that the Cayley graph of G in generators a, b has no dead end vertices. However, if the generating set is A = {a, ab}, then b becomes a dead end element. This example shows that being a dead end element strongly depends on the generating set. However, if the generating set is clear, we will often say that an element of G is (or is not) a dead end. A less trivial example of dead end elements comes from the R.Thompson group F . Recall that F is the group presented by 〈 x0, x1, x2, . . . | xjxi = xixj+1 (j > i) 〉. This group can be generated by x0, x1. In these generators, the group has infinitely many dead end elements. Say, g = x0x1x6x −1 3 x −2 0 will be a dead end element of length 11. One can see details in [11]. (There are no dead end elements in F of length less than 11.) There are descriptions of all dead end elements in F given independently in [2] 2000 Mathematics Subject Classification. Primary 20F32; Secondary 05C25.