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• W37564491 abstract "Let V (≥ F′) be a fully invariant subgroup of a free group F = 〈x1,…, x n 〉 of rank n ≥ 2. A system w(V) = (w1 V,…, w m V), m ≤ n, of cosets of F / V is said to be primitive in F / V if it can be extended to a basis of F / V; and if w(V) is a primitive system in F / V then w(V) is said to lift to a primitive system of the free group F if there exists v = (v1,…, v m ), v i ∈ V, such that the system w(v) = (w1v1, …. , w m v m ) is primitive in F. Primitivity of a system w(V) = (w1 V,…, w n V), with n = rank (F), defines an automorphism α : F / V → F / V mapping x i V → w i V, i = 1,…, n. Clearly, every automorphism, α : F → F induces an automorphism α : F / V → F / V (α ∈ Aut (F / V)). An automorphism α ∈ Aut (F / V) is said to be tame if it is induced by an automorphism α : F → F (α ∈ Aut (F)) of the free group; or equivalently, if the corresponding primitive system w(V) = (w1 V,…, w n V) of cosets in F / V lifts to a primitive system w(v) = (w1v1,…, w n v n ) in F. If α ∈ Aut (F / V) is not tame then α may also be called wild or simply non-tame. Thus, for instance, if F is free of rank n ≥ 4, then every automorphism of F / F″ is tame (Bachmuth & Mochizuki, 1985) and consequently, every primitive system mod F″ lifts to a primitive system of F. However, in the case when n = 3, while the primitive system (x[y, z, x, x], y, z) mod F″ does not lift to a primitve system of F (Chein, 1968); the primitive element x[y, z, x, x]F″ is the same as x[x, [y, z, x]−1]F″ which in turn lifts to a part of the basis {x[x, [y, z, x]−1], y[y, [y, z, x]−1], z[z, [y, z, x]−1]} of F." @default.
• W37564491 created "2016-06-24" @default.
• W37564491 creator A5041935403 @default.
• W37564491 date "1999-01-01" @default.
• W37564491 modified "2023-09-23" @default.
• W37564491 title "Around Automorphisms of Relatively Free Groups" @default.
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• W37564491 doi "https://doi.org/10.1007/978-3-0348-9996-3_4" @default.
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