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Pseudocharacters and the Problem of Expressibility for Some Groups

✍ Scribed by Valeriĭ A. Faĭziev; Prasanna K. Sahoo

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
247 KB
Volume
250
Category
Article
ISSN
0021-8693

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✦ Synopsis

For a class of extensions of free products of groups, a description is given of the space of the real-valued functions ϕ defined on the group G and satisfying the conditions (1) the set ϕ xy -ϕ x -ϕ y x y ∈ G is bounded; and (2) ϕ x n = nϕ x for any x ∈ G and any n ∈ (the set of integers).

Let G be an arbitrary group and let S be its subset such that S -1 = S. Suppose gr S is the subgroup of G generated by S. Denote by l S g the length of an element g ∈ gr S relative to the set S. Let V be a finite subset of a free group F of countable rank, and let the verbal subgroup V F be a proper subgroup of F. For arbitrary group G, denote by V G the set of values in the group G of all the words from the set V . This paper establishes the infinity of the set l S g g ∈ V G , where G belongs to a class of extension of free products of groups, S = V G ∪ V G -1 .

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The work for this paper was carried out partly at the Courant Institute of Mathematical Sciences under NSF Grant GP-12024. Reproduction in whole or in part is permitted for any purpose of the United States Government. Communicated through G. Baumslag.