Résume | A finitely generated group G has unbounded depth with respect to a finite generating set S if for any n≥1, there exists g in G such that no element of the form gh, with h of word length at most n, has larger word length than g. In other words, g locally maximizes the word length in its n-neighborhood. The existence of infinite groups with unbounded depth is not evident, and the first example of such a group was provided by Cleary and Taback, who showed that the lamplighter group Z/2Z wr Z has unbounded depth for a standard generating set.
In this talk we will concentrate on the case where G=A wr B is the wreath product of two groups. An essential tool for us is the description of geodesics on A wr B in terms of the solutions to the Traveling Salesman Problem on B, due to Parry. We prove that for any finite group A and any finitely generated group B, the group A wr B admits a standard generating set with unbounded depth, and that if B is abelian then the above is true for every standard generating set. When B = H * K is the free product of two finite groups H and K, we characterize which standard generators of the associated wreath product have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. In particular, our result shows a difference with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups does not have unbounded depth with respect to some standard generating set. |