Gröbner-Shirshov bases for some one-relator groups. (Q2880067)

Some one relator groups were studied be means of groups with standard normal forms (the standard GS bases) in the sense of \textit{L. A. Bokut'}, [Algebra Logika 5, No. 5, 5-23 (1966; Zbl 0189.01103); ibid. 6, No. 1, 15-24 (1967; Zbl 0189.01201)]. It is known that any 1-relator group can be can effectively embedded into a 2-generator 1-relator group \(G=gp(x,y\mid x^{i_1}y^{j_1}\cdots x^{i_k}y^{j_k},\;k\geq 1)\), \(k\) is the depth. It is proved that a group \(G\) of \(\text{depth}\leq 3\) is computably embeddable in a Magnus tower of HNN-extensions with standard normal form of elements. There are quite a lot of examples that support the old conjecture that the result is valid for any depth.