On the algebraic classification of subgroups of hyperbolic planar crystallographic groups

Costa, Antonio F. (2018) On the algebraic classification of subgroups of hyperbolic planar crystallographic groups. In: Algebraic curves and their applications. Contemporary Math., 724 . American Mathematical Society, USA, pp. 225-236. (In Press)

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Abstract

A planar Euclidean or hyperbolic crystallographic group $\Delta$ is a subgroup of the group of isometries of the Euclidean plane $\mathbb{E}^{2}$, respectively the hyperbolic plane $\mathbb{H}^{2}$, with compact orbit space. These groups are classified algebraically by a symbol called signature and an equivalence relation defined on the set of signatures. In 1990 A.H.M. Hoare gave an algorithm to obtain the signature of a finite index subgroup of a planar crystallographic group. Recently the authors completed the algorithm of Hoare and implemented it on a computer system. In the signature of the hyperbolic groups there is a sign $+\,$or $-$, in the case of $+$ sign the cyclic order on some integers in signatures are essential to determine the isomorphism class. In this article we show examples where such cyclic order is necessary to determine if two subgroups of a given hyperbolic group are isomorphic. Finally we announce the implementation of the algorithm to compute the signature of subgroups of hyperbolic crystallographic groups on the computer system for group theory GAP.

Item Type: Book Section
Subjects: Q Science >
Divisions: Engineering, Science and Mathematics
Engineering, Science and Mathematics > Mathematics
Depositing User: Editor Risat
Date Deposited: 03 Oct 2018 01:40
Last Modified: 03 Oct 2018 01:40
URI: http://repo.risat.org/id/eprint/356

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