On the torsion in a group \bf F/[M,N] in the case of combinatorial asphericity of groups \bf F/M and \bf F/N

2024/02/16 by Olga V. Kulikova, Kulikova, O. V.
Computer Science · Mathematics · #20F05 #20F06 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2402.10531

openalex publication_date 2024/02/16 · openalex created_date 2024/02/20 · openalex updated_date 2026/07/28

Abstract

Let F be a non-Abelian free group with basis A, M and N be the normal closures of sets R_M and R_N of words in the alphabet A^{± 1}. As is known, the group F/[N, N] is torsion-free, but, in general, torsion in F/[M, N] is possible. In the paper of Hartley and Kuz'min (1991), it was proved that if R_M=\v\, R_N=\w\ and words v and w are not a proper power in F, then F/[M,N] is torsion-free. In the present paper a sufficient condition for the absence of torsion in F/[M,N] is obtained, which allows to generalize the result of Hartley and Kuz'min to arbitrary words v and w.

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