Quotient groups of IA-automorphisms of free metabelian groups

2020/12/24 by C. E. Kofinas, Kofinas, C. E., A. I. Papistas +1
Computer Science · Mathematics · #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2012.13286

openalex publication_date 2020/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a positive integer n, with n ≥ 2, let M_n be a free metabelian group of rank n. For c ∈ ℕ, let γ_c(M_n) be the c-th term of the lower central series of M_n. For c ≥ 2, let \rm I_c\rm A(M_n) be the subgroup of \rm Aut(M_n) consisting of all automorphisms inducing the identity mapping on M_n/γ_c(M_n). In this paper, we study the quotient groups \cal L^c(\rm IA(M_n)) = \rm I_c\rm A(M_n)/\rm I_c+1\rm A(M_n) for all n and c. For c ≥ 2, we show γ_c(\rm IA(M₂)) = \rm I_c+1\rm A(M₂)). For n = 3, we show γ₃(\rm IA(M₃)) ≠ \rm I₄\rm A(M₃) and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for n = 3 and c = 3. For n ≥ 4 and c ≥ 3, we prove that \cal L^c(\rm IA(M_n)) = γ_c-1(\rm IA(M_n))\rm I_c+1\rm A(M_n)/\rm I_c+1\rm A(M_n).

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