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A problem of expressibility in some amalgamated products of groups
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 71 / Issue 1 / August 2001
- Published online by Cambridge University Press:
- 09 April 2009, pp. 105-115
- Print publication:
- August 2001
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- Article
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Let S be a subset of a group G such that S−1 = S. Denote by gr (S) the subgroup of G generated by S, and by ls(g) the length of an element g ∈ gr(S) relative to the set S. Suppose that V is a finite subset of a free group F of countable rank such that the verbal subgroup V (F) is a proper subgroup of F. For an arbitrary group G, denote by
(G) the set of values in G of all the words from the set V. In the present paper, for amalgamated products G = A *HB such that A ≠ H and the number of double cosets of B by H is at least three, the infiniteness of the set {ls(g) | g ∈ gr(S)}, where S = (G) ∪ (G)−1, is estabilished.
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