For what sequences {an} of points of the open unit disc D does there exist a constant k such that

for all bounded harmonic functions f on D?

A k-tuple of members of a group is free if it freely generates a free group. In an earlier paper [1] the authors have shown:

The generalization of the Magnus embedding [7] proved by Smelkin [9] may bestated as follows. Let L be a free group freely generated by the set xi(i∈I), and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and ∏ is the V-freegroup with free generating set the symbols [g, xi] (g∈G, i∈I), then L/V(R) is embeddedin the semidirect product ∏ ⋊ G (where the action of G on ∏ is given by h · [g, xi] = [hg, xi], for h, g ∈ G).

The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentation

the subgroup PSL(2, ℤ) being generated by UV and VW.

Let K be a field, G a finite group. Let V be an (irreducible) KG-module, where KG is the group algebra consisting of all formal sums . The action of on α = ∑aθg on element ν ∈ V obeys the rule If H is a subgroup of G, then, restricting the action of G on V to H, V is also a KH-module. Notation: VH.