On the torsion in a group $\bf F/[M,N]$ in the case of combinatorial asphericity of groups $\bf F/M$ and $\bf F/N$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916128208453632 |
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| author | Kulikova, O. V. |
| author_facet | Kulikova, O. V. |
| contents | 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 $A^{\pm 1}$. 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 $w$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10531 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the torsion in a group $\bf F/[M,N]$ in the case of combinatorial asphericity of groups $\bf F/M$ and $\bf F/N$ Kulikova, O. V. Group Theory 20F05, 20F06 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 $A^{\pm 1}$. 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 $w$. |
| title | On the torsion in a group $\bf F/[M,N]$ in the case of combinatorial asphericity of groups $\bf F/M$ and $\bf F/N$ |
| topic | Group Theory 20F05, 20F06 |
| url | https://arxiv.org/abs/2402.10531 |