Normal subgroups of even index in an NEC group
We analyze normal subgroups of a NEC group \(\Gamma\) (discrete subgroup of the isometry group of the non-euclidean plane U, with compact quotient space) when the index of the subgroup with respect to the group is even. The technique we use is the study of a fundamental region for the subgroup derived from a fundamental region for the group. In Sect. 2 we give definitions and basic results. In Sect. 3 we compute the period cycles in the signature of a normal subgroup of even index in a NEC group. As a consequence of these results, in Sect. 4 we establish conditions on the signatures of the groups for a NEC group to have a normal subgroup of even index which is a surface group. This allows us to find the signatures of groups \(\Gamma\) and \(\Gamma\) ' for which \(G\cong \Gamma '/\Gamma\) is an automorphism group of even index of the Klein surface, and this to find signatures of NEC groups \(\Gamma\) for which U/\(\Gamma\) is a Klein surface having an automorphism of order 2.
scientific article; zbMATH DE number 4000216
scientific article