ONE-SIDE NIELSEN TRANSFORMATIONS IN FREE GROUPS

We study a graph related to the Andrews—Curtis graph. The vertices are pairs of elements of a free group, and two vertices are adjacent if they can be obtained from one another by a left Nielsen transformation, that is, for each two words u, v the pair (u, v) is adjacent to (vu, v), (v‑1u, v), (u, uv), and (u, u‑1v). We study connected components of the graph and describe the shortest pairs in all components. Hence, we find a linear-time algorithm for checking whether two pairs belong to the same connected component of the graph. </jats:p