Parametrization of reflection groups acting in a disk (Q1824726)

This paper deals with the problem of finding a representation of the space of embeddings of a group of Fuchsian type in PSL(2,\({\mathbb{R}})\). Despite the title, which refers to NEC groups, the major results are given for orientation-\(preserving\) groups. The authors give a method of deducing the general case from this one. It has been known at least from the treatise of Fricke-Klein that the traces of finitely many group elements suffice to embed this space in \({\mathbb{R}}^ N\) and it can be given as a semi-algebraic set (Helling). Here the authors analyze the question in much more detail especially for surface groups of genus \(p>1\). They first use 6p-4 traces to give the embedding so that the image is of codimension 2. They then introduce a discrete parameter which they exploit to give an embedding into \({\mathbb{R}}^{6p-6}\) (i.e. a ``complete set of moduli'').