Computing the signatures of subgroups of non-Euclidean crystallographic groups
Cortázar, Ismael y Costa González, Antonio F. . (2017) Computing the signatures of subgroups of non-Euclidean crystallographic groups. Journal of Algebra, 477, 483-493
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A (planar and cocompact) non-Euclidean crystallographic (NEC) group is a subgroup of the group of (conformal and anti-conformal) isometries of the hyperbolic plane H2 such that H2= is compact. NEC groups are classifed alge- braically by a symbol called signature. In this symbol there is a sign + or and, in the case of sign +, some cycles of integers called period-cycles have an essential direction. In 1990 A.H.M. Hoare gives an algorithm to obtain the signature of a nite 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.
