Algorithmic undecidability of compatibility problem for equations in free groups: explicit equations with one commutator-type constraint
Let \(F_{2}=\langle a,b \rangle\) be a free group with free generators \(a\), \(b\) and let \(F_{2}^{(s)}\) be the \(s\)-th term of the derived series of \(F_{2}\). The main results proved in this paper are. Theorem 1. In \(F_{2}\) one can construct an equation \(w(x,x_{1},x_{2}, \ldots, x_{n},a,b)=1\) with unknowns \(x_{1}, x_{2}, \ldots, x_{n}\) and parameter \(x\), such that there exists no algorithm to recognize for an arbitrary \(k \in \mathbb{N}\), whether the equation \[w(a^{k},x_{1},x_{2}, \ldots, x_{n},a,b)=1\] has a solution satisfying the condition \(x_{1},x_{2}, \ldots, x_{t} \in F_{2}^{(1)}\) where \(t\) is some fixed integer between \(1\) and \(n\). Theorem 3. There exists no algorithm to recognize, whether an arbitrary equation \(w(x_{1},x_{2}, \ldots, x_{n})=[a,b]\) in the free group \(F_{2}=\langle a,b \rangle\) has a solution \(g_{1}, g_{2}, \ldots, g_{n}\) satisfying the condition \(g_{1} \in F_{2}^{(2)}\).
scientific article; zbMATH DE number 7290698
scientific article