Discontinuous subgroups of PGL\(_{2}(K)\). (Q1421802)

Let \(K\) be a non-archimedean valued field of characteristic \(p\geq 0\) and with residue field \(k\) of characteristic \(p_k \geq 0\). The authors give a careful analysis of what are the possible finitely generated discontinuous subgroups \(\Gamma\) of \(PGL(2,K)\) that can act on \(\mathbb{P}^1(K)\) minus its set of limit points with quotient \(\mathbb{P}^1\). A free normal finite index subgroup of such a \(\Gamma\) gives rise to a finite branched cover of the projective line by a Mumford curve. The authors determine the possible structure of such \(\Gamma\) as amalgams of abstract finite groups and give a formula for the number of branch points in the corresponding cover (except if \(p=0\) and \(p_k \leq 5\)). This includes settling the question which (abstract) amalgams of groups are actually realizable as such \(\Gamma\) (compare: in \textit{G.~Cornelissen, F.~Kato} and \textit{A.~Kontogeorgis} [Math.\ Ann. 320, No. 1, 55--85 (2001; Zbl 1031.14011)] is a classification of abstract types of groups for \(p>0\) with \(\leq 3\) branch points, without addressing the realizability question).