On Infinite Discontinuous Groups

Abstract

The general discontinuous group is given by n generators and m relations between them

$$ \begin{array}{*{20}c} {\text{R}_\text{1} \text{(S}_{\text{i}_\text{1} } \ldots ) = 1} \\ { \ldots \ldots \ldots \ldots \ldots } \\ {\text{R}_\text{m} \text{(S}_{\text{i}_\text{m} } \ldots ) = 1,} \\ \end{array} $$

as first defined by Dyck (Math. Ann., 20 and 22). The results of those works, however, relate essentially to finite groups. The general theory of groups defined in this way at present appears very undeveloped in the infinite case. Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.