Rank 1 character varieties of finitely presented groups (Q684199)

This paper under review deals with rank 1 character varieties of finitely presented groups. In this article, \(\Gamma\) is a finitely presentable discrete group and \(\chi(\Gamma ,G)\) is the \(G\) character variety of \(\Gamma\) where \(G\) is a rank 1 complex affine algebraic group. An effective algorithm is presented in order to determine the structure of the variety \(\chi(\Gamma, G)\). More precisely, the proposed algorithm takes a finite presentation for \(\Gamma\) and produces a finite presentation of the coordinate ring of \(\chi(\Gamma,G)\). The implementations in Mathematica, as well in Python and SageMath are given. Known theorems of Vogt, Horowitz, Magnus, Gonzalez-Acuna and Montesinos-Amilibia, Brumfiel-Hilden and Drensky are used for the construction of the algorithm. Assuming that \(G=\mathrm{SL}_2(\mathbb{C})\), it provides a new proof and a new description of the structure of \(\mathrm{SL}_2\) character varieties. In the last section of this paper, concrete examples are provided and it is shown that using this algorithm, known results are verified.