Equations over groups. (Q2874364)

Let \(G_X=F(X)*G\) be the free product of the free group \(F(X)\) over the alphabet \(X\) and the group \(G\). Let \(u(x_1,\ldots,x_k)=g_0x_{i_1}^{\varepsilon_1}g_1\cdots g_{n-1}x_{i_n}^{\varepsilon_n}\in G_X\), where each \(g_j\) is a group element of \(G\), each exponent \(\varepsilon_j\) is \(1\) or \(-1\) and each \(x_{i_j}\) is taken from the alphabet \(X=\{x_1,\ldots,x_k,\ldots\}\). An equation in \(k\) variables over the group \(G\) is an expression of the form \(u(x_1,\ldots,x_k)=1\). The \(g_j\) are the coefficients (or constants) of the equation and \(n\) is the length of the equation. A solution of \(u(x_1,\ldots,x_k)=1\) in the group \(G\) is an assignment \(x_j\to h_j\in G\) such that \(g_0h_{i_1}^{\varepsilon_1}g_1\cdots g_{n-1}h_{i_n}^{\varepsilon_n}=1\).NEWLINENEWLINE It may be that the equation \(u(x_1,\ldots,x_k)=1\) has no solution in the group \(G\), but it may have a solution in an overgroup \(H\) of \(G\), then it is said that this equation is solvable over the group \(G\). This is equivalent to that the canonical map from \(G\) to the group \(H=G_X/U\) is an embedding, where \(U\) is the normal closure of \(u(x_1,\ldots,x_k)\) in \(G_X\).NEWLINENEWLINE The process of solving equations is central not only in group theory, but in other branches of mathematics. In general there are two questions to answer. The first is about the existence of a solution in a group (or in an overgroup), the second one is about finding a solution if one is known to exist.NEWLINENEWLINE There is a long story in the research about the answer to the above (and related) questions according to the variety where the group \(G\) belongs to.NEWLINENEWLINE In this survey paper the author makes a comprehensive exposition of all the progress from the beginning until recently. His makes comments and remarks, with a detailed list of references, guiding the interested reader in this area of group theory and facing him with open problems.