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

O. V. Kulikova

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

O. V. Kulikova

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 $A^{\pm 1}$. 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 $w$.

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

Abstract

Let

be a non-Abelian free group with basis

and

be the normal closures of sets

and

of words in the alphabet

. As is known, the group

is torsion-free, but, in general, torsion in

is possible. In the paper of Hartley and Kuz'min (1991), it was proved that if

and words

and

are not a proper power in

, then

is torsion-free. In the present paper a sufficient condition for the absence of torsion in

is obtained, which allows to generalize the result of Hartley and Kuz'min to arbitrary words

and

Paper Structure (10 theorems, 5 equations)

This paper contains 10 theorems, 5 equations.

Key Result

Lemma 1

A word $w$ in the alphabet $A^{\pm 1}$ represents the identity of the group $G$ defined by $\mathcal{P} = \langle A\mid R\rangle$ if and only if there is a based picture $\mathbf{P}$ over $\mathcal{P}$ with boundary label identically equal to $w$.

Theorems & Definitions (10)

Lemma 1
Lemma 2
Theorem 1
Lemma 3
Corollary 1
Corollary 2
Corollary 3
Lemma 4
Corollary 4
Theorem 2