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On the group of reduced identities of relatively free solvable groups. (Russian, English) Zbl 1020.20019
Sib. Mat. Zh. 43, No. 5, 1142-1148 (2002); translation in Sib. Math. J. 43, No. 5, 920-925 (2002).
The author uses the basic notions of algebraic geometry over a fixed group $$G$$ [see G. Baumslag, A. Myasnikov, V. Remeslennikov, J. Algebra 219, No. 1, 16-79 (1999; Zbl 0938.20020)]. A group $$H$$ with fixed embedding $$G\to H$$ is called $$G$$-group. Denote the free product $$G*F(X)$$ by $$G[X]$$. An element $$v(g_1,\dots,g_m,x_1,\dots,x_n)\in G[X]$$ is called $$G$$-identity if $$v(g_1,\dots,g_m,h_1,\dots,h_n)=1$$ for all $$h_1,\dots,h_n\in H$$. Let $$V(G)$$ be the verbal subgroup of $$G[X]$$ corresponding to the group of all coefficient-free identities, and let $$V_c(G)$$ be the verbal subgroup of $$G[X]$$ corresponding to the group of all $$G$$-identities. The quotient group $$V_{n,\text{red}}(G)$$ of the group of all elements of $$V_c(G)$$ of $$n$$ variables modulo the group of all elements of $$V(G)$$ of $$n$$ variables is called the group of reduced identities of rank $$n$$.
The main result is that the groups $$V_{n,\text{red}}(G)$$, $$n\geq 1$$, are trivial for finitely generated free solvable groups and free metabelian groups of a given class.
##### MSC:
 20E10 Quasivarieties and varieties of groups 14A22 Noncommutative algebraic geometry 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07 Subgroup theorems; subgroup growth 20E34 General structure theorems for groups 20F65 Geometric group theory 20F16 Solvable groups, supersolvable groups 20F18 Nilpotent groups
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