Computing the signatures of subgroups of non-Euclidean crystallographic groups

A (planar and cocompact) non-Euclidean crystallographic (NEC) group Δ is a subgroup of the group of, conformal and anti-conformal, isometries of the hyperbolic plane H2 such that H2/Δ is compact. NEC groups are classified algebraically by a symbol called signature. In this symbol there is a sign + or − and, in the case of sign +, some cycles of integers in the signature, called period-cycles, have an essential direction. In 1990 A.H.M. Hoare gave an algorithm to obtain the signature of a finite index subgroup of an NEC group. The process of Hoare fails in some cases in the task of computing the direction of period-cycles. In this work we complete the algorithm of Hoare, this allows us to construct a program for computing the signature of subgroups of NEC groups in all cases.