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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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