The universal torsion-free image of a group
Let \(\rho(G)=\bigcap\{N\triangleleft G\), \(G/N\) torsion free\} be the torsion-free radical of a group \(G\), \(\widehat G=G/\rho(G)\) be the torsion-free image of \(G\). Theorem 1.4. Let \(G\) be a group with a presentation having at most 5 relators, each of length at most 3. Then \(\widehat G\) is locally indicable. Theorem 1.6. Let \(A\), \(B\) be torsion-free groups, and \(w\in A*B\) a cyclically reduced word of length at least 2. Suppose \(a\in A\) can be expressed in the form \(a=\prod^n_{i=1}v_iw^{\varepsilon(i)}v_i^{-1}\) with \(n\leq 4\) and \(\varepsilon(i)=\pm1\) for each \(i\). Then \(a=1\). F. Levin's conjecture (1962) is equivalent to the assertion of this theorem, without the restriction on \(n\).
scientific article; zbMATH DE number 1033786
scientific article