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Project: cocalc-sagemath-dev-slelievre
Views: 418346����; � TeX output 2012.05.29:1507� ������� ��p ����� �Y��N�e9V $ phvb7t�NAc��G�lib��<��� � |��KǍ�Y��A� �GAP4�P���ac��G�ka���g�\%e��Aܖ�� ��'�P�e9V phvb7t�PComputations��&with��^��� ��Almost��&Cr�!Fystallographic�Gr��uoups��<���� �fb��uy��<� �� ��"Karel��&Dekimpe�!F,�� ��Y)^�;3{� ptmr7t�Katholiek��ge�� Uni�� v�٠ersiteit�Leuv�en�Campus�K��gortrijk,�Uni�� v�ersitaire�Campus,���� �,B-8500�� K��gortrijk,�Belgium��;j��� ��PBettina��&Eic��uk,���� ��n�Institut�� f�� �Ğ�3�ur�Geometrie,�Uni�� v�٠ersit�� ���3�at�Braunschweig,���� ���38106�� Braunschweig,�German�٠y��� �� *��� ��p ���Լ\����8P�NContents����N ������ �H�ߌ� ptmb7t�H1���$ The�� Almost�Crystallographic�Gr���oups�P��gackage��� ��3������ ������� �1.1���$ More�� about�almost�crystallographic�groups��H��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 3������������ �H2���$ Algorithms�� f�� or�almost�crystallographic�gr���oups��� ��5������������� �2.1���$ Properties�� of�almost�crystallographic�groups��Y��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 5������������� 2.2���$ Betti�� numbers��\��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 5������������� 2.3���$ Determination�� of�certain�e�٠xtensions� )��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 6������������ �H3���$ The�� catalog�of�almost�crystallographic�gr���oups��� ��m7������������� �3.1���$ Rational�� matrix�groups������ .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 7������������� 3.2���$ Polyc�٠yclically�� presented�groups�d.��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 8������������� 3.3���$ More�� about�the�type�and�the�dening�parameters� �/��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 9������������� 3.4���$ The�� electronic�v�٠ersus�the�printed�library��h��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 10������������ �H4���$ Example�� computations�with�almost�crystallographic�gr���oups��� �712������������� �4.1���$ Example�� computations�I� ;&��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 12������������� 4.2���$ Example�� computations�II�殍�� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 13������������� 4.3���$ Example�� computations�III��6��� .�� ��� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .����� .���� 13����������$ �HBibliograph�٠y���nC16����������$ Index������17������� �� ���� ��p ���Ƣ�������M�e9V ` phvb7t�M1���� �� E�NThe��Almost��, �� ��>Cr�\%ystallographic���� ��Gr��G�oups���P���ac�ka���g�\%e����<ٍ�A�]�group�]�is�]�called��Halmost�crystallographic��if�it�is�a�nitely�generated�nilpotent-by-nite�group�without�non-tri�� vial�� �nite���normal�subgroups.�An���important�special�case�of�almost�crystallographic�groups�are�the��Halmost�Bieberbach���gr���oups�:�� these�are�almost�crystallographic�and�torsion�free.��N8�By��its�denition,��an�almost�crystallographic�group��LKj� ptmri7t�LG��has�a�nitely�generated�nilpotent�normal�subgroup��LN��D�of�nite���inde�٠x.���Clearly��Y�,����LN�N#�is�polyc�yclic�and���thus�has�a�polyc�٠yclic�series.�The�number�of�innite�c�٠yclic�f��gactors�in�such�a�series���for�� �LN���is�an�in���v�� ariant�of��LG�:�the��HHirsch�length��of��LG�.���F�٠or�#<each�#;almost�crystallographic�group�of�Hirsch�length�3�and�4�there�e�٠xists�a�representation�as�a�rational�matrix�group���in�{dimension�4�or�5,�respecti�� v�٠ely��Y�.�These�representations�can�{be�considered�as�ane�representations�of�dimension�3�or���4.��V��fgia��these�representations,�the�almost�crystallographic�groups�act�(properly�discontinuously)�on��5��� msbm10�R���^��?�V zptmcm7t�3����or��R���^��4�� �.�That�is���one�� reason�to�dene�the��Hdimension��of�an�almost�crystallographic�group�as�its�Hirsch�length.���The�F3-dimensional�Gand�a�part�of�the�4-dimensional�almost�crystallographic�groups�ha���v�٠e�been�classied�by�K.�Dekimpe���in��,[Dek96].��-This�classication�includes�all�almost�Bieberbach�groups�in�dimension�3�and�4.�It�is�the�rst�central�aim���of�;�this�;�package�to�gi�� v�٠e�access�to�the�resulting�library�of�groups.�The�groups�in�this�electronic�catalog�are�a���v�� ailable���in��tw��go��dierent�representations:�as�rational�matrix�groups�and�as�polyc�٠yclically�presented�groups.�While�the�rst���representation��\is�the�more�natural�one,�the�latter��[description�f��gacilitates�eecti�� v�٠e�computations�with�the�considered���groups�� using�the�methods�of�the��Q�l� phvr7t�QP���olycyclic��package.���The�d�second�aim�of�this�package�is�to�introduce�a�v�� ariety�of�algorithms�for�computations�with�polyc�٠yclically�presented���almost�<Zcrystallographic�<[groups.�These�algorithms�supplement�the�methods�a���v�� ailable�in�the��QP���olycyclic��package�and���gi�� v�٠e� zaccess�to�some� {methods�which�are�interesting�specically�for�almost�crystallographic�groups.�In�particular���,�we���present�U�methods�to�compute�Betti�U�numbers�and�to�construct�or�check�the�e�٠xistence�of�certain�e�٠xtensions�of�almost���crystallographic�V�groups.�W��37e�note�V�that�these�methods�ha���v�٠e�been�applied�in�[DE01b]�and�[DE01a]�for�computations���with�� almost�crystallographic�groups.���Finally��Y�,���we�remark�that�almost�crystallographic�groups�can���be�seen�as�natural�generalizations�of�crystallographic���groups.��7A��,library�of��8crystallographic�groups�and�algorithms�to�compute�with�crystallographic�groups�are�a���v�� ailable�in���the�� �QGAP��packages����<x cmtt10�cryst�,��carat��and��crystcat�.��N8��P1.1�� More��&about�almost�cr�!Fystallographic�gr��uoups��N8��Almost��crystallographic�groups�were�rst�discussed�in�the�theory�of�actions�on�Lie�groups.�W��37e�recall�the�original���denition�� here�brie y�and�we�refer�to�[Aus60],�[Dek96]�and�[Lee88]�for�more�details.���Let��%�LL����be��$a�connected�and�simply�connected�nilpotent�Lie�group.�F�٠or�e�xample,�the�3-dimensional��$Heisenber���g�group,���consisting�d�of�d�all�upper�unitriangular��>�V zptmcm7t�3�ӆ�Dƛ� zptmcm7y�D��3�{matrices�with�d�real�entries�is�of�this�type.�Then��LL��-�o��ӆ�Aut���}(�LL���)��acts�anely�(on���the�� left)�on��LL����via���� �YG�D8�Ll���A���? zptmcm7m�A;����Ll�������Eƛ� zptmcm7y�E0��6l�D2���LL���A;��D8�A����D2���Aut���(�LL���)�:���0�����(�TKj� ptmri7t�Tl���B���? zptmcm7m�B;� ���)��`}�Ll��������E0��6l�=��Ll��0�A� �x�(�Ll��������E0��oT�)�D��� �� ���� ��4� �l�LChapter�� 1.�The�Almost�Crystallo��ggr�٠aphic�Gr���oups�P��37ac���ka�g�e���p ���� ��Let�B�LC�zV�be�a�maximal�compact�subgroup�of��B�Aut��A�(�LL���)�.�Then�a�subgroup��LG��of��LL�Z��o�SU�LC��is�said�to�be�an�almost�crystallographic�� �group�\]if�and�only�if�the�action�of��LG�\^�on��LL���,�induced�by�the�action�of��LL����o����Aut���(�LL��)�,�is�properly�discontinuous�and�the���quotient���space��LG�Dn�LL����is�compact.�One�reco�٠v�ers���the���situation�of�the�ordinary�crystallographic�groups�by�taking��LL���=���R���^��Tn�� �,���for�� some��Ln�,�and��LC��h�=���LO�(�Ln�)�,�the�orthogonal�group.��N8�More�;generally��Y�,�we�say�that�an�abstract�group�is�an�almost�crystallographic�group�if�it�can�be�realized�as�a�genuine���almost��crystallographic��subgroup�of�some��LL� ���o� �(�LC�8P�.�In�the�follo�� wing�theorem�we�outline�some�algebraic�characterizations���of��Falmost�crystallographic��Ggroups;�see�Theorem�3.1.3�of�[Dek96].�Recall�that�the��HFitting�subgr���oup�Fitt�(�LG�)��of�a���polyc�٠yclic-by-nite�� group��LG��is�its�unique�maximal�normal�nilpotent�subgroup.��N8��HTheor���em.� �K3{� ptmro7t�KThe�� follo�� wing�are�equi�v�alent�for�a�polyc�٠yclic-by-nite�group��LG�K:����� W(1)�� �LG�� �Kis�an�almost�crystallographic�group.����� W(2)�� Fitt�(�LG�)�� �Kis�torsion�free�and�of�nite�inde�٠x�in��LG�K.����� W(3)�� �LG�� �Kcontains�a�torsion�free�nilpotent�normal�subgroup��LN���Kof�nite�inde�٠x�in��LG��Kwith��LC����TG�����(�LN� ���)��Ktorsion�free.����� W(4)�� �LG�� �Khas�a�nilpotent�subgroup�of�nite�inde�٠x�and�there�are�no�non-tri�� vial�nite�normal�subgroups�in��LG�K.����In�� particular���,�if��LG��is�almost�crystallographic,�then��LG�A=�LF���itt�.�(�LG�)��is�nite.�This�f��gactor�is�called�the��Hholonomy�gr���oup��of��LG�.���The��Jdimension��Iof�an�almost�crystallographic�group�equals�the�dimension�of�the�Lie�group��LL� ��abo�٠v�e�which��Jcoincides�� �also�ˆwith�˅the�Hirsch�length�of�the�polyc�٠yclic-by-nite�group.�This�library�therefore�contains�f��gamilies�of�virtually���nilpotent�� groups�of�Hirsch�length�3�and�4.��� �� ��� ��p ���C�������M2���� ��^N��NAlgorithms� 2f��G�or�almost��, �� ��!cr�\%ystallographic����Z�qgr��G�oups����;���This��jchapter��ipresents�a�v�� ariety�of�algorithms�for�almost�crystallographic�groups.�In�most�cases,�the�٠y�assume�a�poly-�� �c�٠yclically��presented�group�as�input;��in�particular���,�the�input�groups�must�be�polyc�yclic��in�this�case.�The�methods���described��Yhere��Xsupplement�the�methods�of�the��QP���olycyclic��package�for�polyc�٠yclically�presented�groups.�Man�٠y�of�the���functions�7in�6this�chapter�are�based�on�methods�of�the��QP���olycyclic��package�and�thus�this�package�must�be�installed�to�use���the�WIfunctions�introduced�WHhere.�W��37e�refer�to�the��QP���olycyclic��package�for�further�information�on�polyc�٠yclic�presentations.��N8��P2.1�� Pr��uoper�B�ties��&of�almost�cr�!Fystallographic�gr�oups�� �����3�S3{� ptmr7t�S1���̟��^�3�u�7 msam7�I���IsAlmostCrystallographic(�?��LG��)�48��P��N8�This���function���checks�if�a�polyc�٠yclically�presented�group��LG��x�is�almost�crystallographic;�that�is,�it�checks�if��LG��x�is�nilpotent-���by-nite�� and�has�no�non-tri�� vial�nite�normal�subgroup.�������3�S2���̟��^�I���IsAlmostBieberbachGroup(�?��LG��)�9x��P��N8�This��function��checks�if�a�polyc�٠yclically�presented�group��LG����is�almost�Bieberbach;�that�is,�it�checks�if��LG����is�nilpotent-���by-nite�� and�torsion�free.��N8��P2.2�� Betti��&n�umber��s��N8��Let�-�LG�-�be�a�polyc�٠yclically�presented�and�torsion�free�group�of�Hirsch�length��Ln�.�Then�we�can�compute�the�Betti�numbers����A����Ti��r(�(�LG�)�p��for�p��Li���D2�f�0�A;����1�A;��2�A;��Ln� ��D ��2�A;����Ln� ��D ��1�A;��Ln�Dg�.�p�If�p��Ln���D��6�,�then�we�p�can�compute�all�Betti�numbers��A����Ti��r(�(�LG�)��for��0���D��Li��D��6��of�p��LG�.�W��37e���introduce�.the�follo�� wing�functions�for�/this�purpose�and�we�refer�to�[Bro82]�for�the�details�on�the�orientation�module���and�� the�Betti�numbers.�������3�S1���̟��^�I���OrientationModule(�?��LG��)�X���F��N8�This� function� determines�the�orientation�module�of�the�polyc�٠yclically�presented�group��LG�;�that�is,�it�returns�a�list�of���matrices�� �Lm����1�� �A;����:�:�:����;����Lm����Tn����D���LGL���(1�A;��Z�)��which�are�the�images�of�the�'Igs(G)'�in�their�action�on�the�orientation�module.�������3�S2���̟��^�I���BettiNumber(�?��LG�,��Lm��)�f�;�F��N8�This��Ifunction�returns�the��J�Lm�th�Betti�number�of�the�polyc�٠yclically�presented�torsion�free�group��LG��-�if��Lm��S�D2�f�0�A;����1�A;��2�A;��Ln����D ����2�A;����Ln�8��D ��1�A;��Ln�Dg�,�� where��Ln��is�the�Hirsch�length�of��LG�.�������3�S3���̟��^�I���BettiNumbers(�?��LG��)�q���A��N8�This��.function��/returns�the�Betti�number�of�the�polyc�٠yclically�presented�torsion�free�group��LG���if�the�Hirsch�length�of��LG����is�� smaller�than�7.��� �� (���� ��6� ��LChapter�� 2.�Algorithms�for�almost�crystallo��ggr�٠aphic�gr���oups���p ���� ��P2.3�� Determination��&of�cer�B�tain�e��xtensions��N8��Let��a�LG��b�be�a�polyc�٠yclically�presented�almost�crystallographic�group.�W��37e�w��gant�to�check�the�e�٠xistence�of�certain�e�٠xten-�� �sions�� of��LG�.��N8�First,�(mit�is�well-kno�� wn�that�(lthe�equi�v�alence�classes�of�e�٠xtensions�of��LG��correspond�(lto�the�second�cohomology�group���of���LG�.�This�cohomology�group�can�be�computed�using�the��methods�of�the��QP���olycyclic��package�for�an�٠y�e�xplicitly�gi�� v�en���module���of��LG�.�Further���,�we���can�construct�a�polyc�٠yclic�presentation�for�each�coc�ycle���of�the�second�cohomology�group.���W��37e�� gi�� v�٠e�an�e�xample�for�such�a�computation�belo�� w��Y�.���Ho�� we�v�٠er���,��we��may�be�interested�in�certain�e�٠xtensions�only;�for�e�٠xample,�the�torsion�free�e�٠xtensions�are�often�of�particu-���lar�W@interest.�If�the�second�W?cohomology�group�is�nite,�then�we�can�compute�a�polyc�٠yclic�presentation�for�each�element���of��athis�group�and�check�the�resulting�group�for�torsion��`freeness.�But�if�the�second�cohomology�group�is�innite,�then���this�L�approach�is�not�a���v�� ailable.�Hence�we�introduce�the�L�follo�wing�special�method�to�co�٠v�er�L�this�and�related�applications.�� �����3�S1���̟��^�I���HasExtensionOfType(�?��LG�,��Ltor��gsionfr��Gee�,��Lminimalcentr�e��)� �� �F��N8�Suppose�C�that��LG�C|�is�a�polyc�٠yclically�presented�almost�crystallographic�group�C�with�Fitting�subgroup��LN� ���.�This�function���checks���if�there���is�a��LG�-module��LM����W����"�D���(��"�=�����P��Z��which�is�centralized�by��LN����such�that�there�e�٠xists�a�torsion�free�e�٠xtension�of��LM����by���LG��g�(if�the� ag��Ltor��gsionfr��Gee��is�true)�or�an�e�٠xtension��LE�-d�with��LZ� � �(�LF���itt�.�(�LE�:��))��(=��LM�X��(if��the�� ag��Lminimalcentr�e��is�true)�or�an���e�٠xtension�� which�satises�both�conditions�(if�both� ags�are�true).���W��37e��note��that�the�e�٠xistence�of�such�e�٠xtensions�is�of�interest�in�the�determination�of�e�٠xtensions�which�are�almost���Bieberbach��groups.��W��37e�refer�to�[DE01b]�for�a�more�detailed�account�of�this�application�and�for�further�results�of�a���similar�� nature.��� �� 4Q��� ��p ���C�������M3���� ��a9��NThe� =�catalog� =�of�almost��, �� ��!cr�\%ystallographic����Z�qgr��G�oups����;�*��This��chapter�introduces��the�access�functions�to�the�catalog�of�3-�and�4-dimensional�crystallographic�groups.�This�� �catalog�� is�an�electronic�v�٠ersion�of�the�classication�obtained�in�[Dek96].��Z��P3.1�� Rational��&matrix�gr��uoups��-��The�0?follo�� wing�three�main�functions�are�a���v�ailable�to�0@access�the�library�of�almost�crystallographic�groups�as�rational���matrix�� groups.���q�����3�S1���̟��^�I���AlmostCrystallographicGroup(�?��Ldim�,��Ltype�,��Lpar�٠ameter��gs��)�����������36�I���AlmostCrystallographicDim3(�?��Ltype�,��Lpar�٠ameter��gs��)�����������36�I���AlmostCrystallographicDim4(�?��Ltype�,��Lpar�٠ameter��gs��)��5���Ldim����is���the�dimension�of�the�required�group.�Thus��Ldim��must�be�either�3�or�4.�The�inputs��Ltype��and��Lpar�٠ameter��gs��are�used���to�Udene�the�desired�group�as�described�in�[Dek96].�W��37e�outline�the�possible�choices�for��Ltype��and��Lpar�٠ameter��gs��here���brie y��Y�.�� A�more�e�٠xtended�description�is�gi�� v�en�later�in�Section�1.1�or�can�be�obtained�from�[Dek96].��=y��Ltype�D��species�the�type�D�of�the�required�group.�There�are�17�types�in�dimension�3�and�95�types�in�dimension�4.�The���input�l�Ltype�l�can�either�be�an�inte�٠ger�dening�the�position�of�the�desired�type�among�all�types;�that�is,�in�this�case��Ltype��is���a��number�in��[1..17]�in�dimension�3�or�a�number�in�[1..95]�in�dimension�4.�Alternati�� v�٠ely��Y�,��Ltype��can�be�a�string�dening���the���desired���type.�In�dimension�3�the�possible�strings�are��"01"�,��"02"�,���A:���:�:���|�,��"17"�.�In�dimension�4�the�possible�strings���are�� listed�in�the�list��ACDim4Types��and�thus�can�be�accessed�from��QGAP�.��=x��Lpar�٠ameter��gs��x�is�a�list�of��yinte�gers.�Its�length�depends�on�the�type�of�the�chosen��ygroup.�The�lists��ACDim3Param��and����ACDim4Param��A�contain��Bat�position��Li��the�length�of�the�parameter�list�for�the�type�number��Li�.�Ev�٠ery�list�of�inte�٠gers�of�this���length�8cis�8da�v�� alid��Lpar�٠ameter�sB�input.�Alternati�v�٠ely��Y�,�8cone�can�input��false��instead�of�a�parameter�list.�Then��QGAP��will�chose���a�� random�parameter�list�of�suitable�length.����9��gap>�?�G�:=�AlmostCrystallographicGroup(�4,�50,�[�1,�-4,�1,�2�]�);����9�<matrix�?�group�of�size�infinity�with�5�generators>����9�gap>�?�DimensionOfMatrixGroup(�G�);����9�5����9�gap>�?�FieldOfMatrixGroup(�G�);����9�Rationals����9�gap>�?�GeneratorsOfGroup(�G�);����9�[�?�[�[�1,�0,�-1/2,�0,�0�],�[�0,�1,�0,�0,�1�],�[�0,�0,�1,�0,�0�],����-��[�?�0,�0,�0,�1,�0�],�[�0,�0,�0,�0,�1�]�],������[�?�[�1,�1/2,�0,�0,�0�],�[�0,�1,�0,�0,�0�],�[�0,�0,�1,�0,�1�],����-��[�?�0,�0,�0,�1,�0�],�[�0,�0,�0,�0,�1�]�],������[�?�[�1,�0,�0,�0,�0�],�[�0,�1,�0,�0,�0�],�[�0,�0,�1,�0,�0�],����-��[�?�0,�0,�0,�1,�1�],�[�0,�0,�0,�0,�1�]�],������[�?�[�1,�0,�0,�0,�1�],�[�0,�1,�0,�0,�0�],�[�0,�0,�1,�0,�0�],��� �� =���� ��8� 讥�LChapter�� 3.�The�catalo��gg�of�almost�crystallo�gr�٠aphic�gr���oups���p ���� ��-���[�?�0,�0,�0,�1,�0�],�[�0,�0,�0,�0,�1�]�],�� ����[�?�[�1,�-4,�1,�0,�1/2�],�[�0,�0,�-1,�0,�0�],�[�0,�1,�0,�0,�0�],����-��[�?�0,�0,�0,�1,�1/4�],�[�0,�0,�0,�0,�1�]�]�]����9�gap>�?�G.1;����9�[�?�[�1,�0,�-1/2,�0,�0�],�[�0,�1,�0,�0,�1�],�[�0,�0,�1,�0,�0�],������[�?�0,�0,�0,�1,�0�],�[�0,�0,�0,�0,�1�]�]����9�gap>�?�ACDim4Types[50];����9�"076"����9�gap>�?�ACDim4Param[50];����9�4����P3.2�� P��z�ol��yc�yc��ulicall�y��&presented�gr�oups��E���All�'the�(almost�crystallographic�groups�considered�in�this�package�are�polyc�٠yclic.�Hence�the�٠y�ha���v�e�a�(polyc�yclic�presen-���tation��Mand��Lthis�can�be�used�to�f��gacilitate�ecient�computations�with�the�groups.�T��37o�obtain�the�polyc�٠yclic�presentation���of���an���almost�crystallographic�group�we�supply�the�follo�� wing�functions.�Note�that�the�share�package��QP���olycyclic��must���be�� installed�to�use�these�functions.���⍍���3�S1���̟��^�I���AlmostCrystallographicPcpGroup(�?��Ldim�,��Ltype�,��Lpar�٠ameter��gs��)�����������36�I���AlmostCrystallographicPcpDim3(�?��Ltype�,��Lpar�٠ameter��gs��)�����������36�I���AlmostCrystallographicPcpDim4(�?��Ltype�,��Lpar�٠ameter��gs��)��G���The� input�is��the�same�as�for�the�corresponding�matrix�group�functions.�The�output�is�a�pcp�group�isomorphic�to�the�cor���-���responding�1Fmatrix�1Ggroup.�An�e�٠xplicit�isomorphism�from�an�almost�crystallographic�matrix�group�to�the�corresponding���pcp�� group�can�be�obtained�by�the�follo�� wing�function.���ፍ���3�S2���̟��^�I���IsomorphismPcpGroup(�?��LG��)��G���W��37e��can�use�the�polyc�٠yclic�presentations�of�almost��crystallographic�groups�to�e�xhibit�structure�information�on�these���groups.���F�٠or���e�xample,�we�can���determine�their�Fitting�subgroup�and�ask�group-theoretic�questions�about�this�nilpotent���group.�ѯThe�f��gactor��LG�A=�LF���it�.�(�LG�)��of�an�almost�crystallographic�group�Ѯ�LG��is�called��Hholonomy�gr���oup�.�W��37e�pro�٠vide�access�to���this�� f��gactor�of�a�pcp�group�via�the�follo�� wing�functions.�Let��LG��be�an�almost�crystallographic�pcp�group.�������3�S3���̟��^�I���HolonomyGroup(�?��LG��)�����������36�I���NaturalHomomorphismOnHolonomyGroup(�?��LG��)��G���The�� follo�� wing�e�٠xample�sho�ws�applications�of�these�functions.����9��gap>�?�G�:=�AlmostCrystallographicPcpGroup(�4,�50,�[�1,�-4,�1,�2�]�);����9�Pcp-group�?�with�orders�[�4,�0,�0,�0,�0�]����9�gap>�?�Cgs(G);����9�[�?�g1,�g2,�g3,�g4,�g5�]�� ��9�gap>�?�F�:=�FittingSubgroup(�G�);����9�Pcp-group�?�with�orders�[�0,�0,�0,�0�]����9�gap>�?�Centre(F);����9�Pcp-group�?�with�orders�[�0,�0�]����9�gap>�?�LowerCentralSeries(F);����9�[�?�Pcp-group�with�orders�[�0,�0,�0,�0�],�Pcp-group�with�orders�[�0�],������Pcp-group�?�with�orders�[� �]�]����9�gap>�?�UpperCentralSeries(F);����9�[�?�Pcp-group�with�orders�[�0,�0,�0,�0�],�Pcp-group�with�orders�[�0,�0�],������Pcp-group�?�with�orders�[� �]�]����9�gap>�?�MinimalGeneratingSet(F);����9�[�?�g2,�g3,�g4�]��� �� I~��� ��LSection�� 3.�Mor��Ge�about�the�type�and�the�dening�par�٠ameter��gs� ��2�9���p ���� ��9��gap>�?�H�:=�HolonomyGroup(�G�);�� ��9�Pcp-group�?�with�orders�[�4�]����9�gap>�?�hom�:=�NaturalHomomorphismOnHolonomyGroup(�G�);����9�[�?�g1,�g2,�g3,�g4,�g5�]�->�[�g1,�identity,�identity,�identity,�identity�]����9�gap>�?�U�:=�Subgroup(�H,�[Pcp(H)[1]^2]�);����9�Pcp-group�?�with�orders�[�2�]����9�gap>�?�PreImage(�hom,�U�);����9�Pcp-group�?�with�orders�[�2,�0,�0,�0,�0�]�����P3.3�� More��&about�the�type�and�the�dening�parameter��s���ٍ�Each���group���from�this�library�kno�� ws�that�it�is�almost�crystallographic�and,�additionally��Y�,�it�kno�� ws�its�type�and�dening���parameters.��������3�S1���̟��^�I���AlmostCrystallographicInfo(�?��LG��)���W��This�\�attrib���ute�\�is�set�for�groups�from�the�library�only��Y�.�It�is�not�possible�at�current�to�determine�the�type�and�the�dening���parameters�� for�an�arbitrary�almost�crystallographic�groups�which�is�not�dened�by�the�library�access�functions.����9��gap>�?�G�:=�AlmostCrystallographicGroup(�4,�70,�false�);����9�<matrix�?�group�of�size�infinity�with�5�generators>����9�gap>�?�IsAlmostCrystallographic(G);����9�true����9�gap>�?�AlmostCrystallographicInfo(G);����9�rec(�?�dim�:=�4,�type�:=�70,�param�:=�[�1,�-4,�1,�2,�-3�]�)��> ��9�gap>�?�G�:=�AlmostCrystallographicPcpGroup(�4,�70,�false�);����9�Pcp-group�?�with�orders�[�6,�0,�0,�0,�0�]����9�gap>�?�IsAlmostCrystallographic(G);����9�true����9�gap>�?�AlmostCrystallographicInfo(G);����9�rec(�?�dim�:=�4,�type�:=�70,�param�:=�[�-3,�2,�5,�1,�0�]�)���ٍ�W��37e�`�consider�the�types�of�`�almost�crystallographic�groups�in�more�detail.�The�almost�crystallographic�groups�in�dimen-���sions�� 3�and�4�f��gall�into�three�f�amilies���W���W(1)�� 3-dimensional�� almost�crystallographic�groups.���ԍ��W(2)�� 4-dimensional�� almost�crystallographic�groups�with�a�Fitting�subgroup�of�class�2.���Ս��W(3)�� 4-dimensional�� almost�crystallographic�groups�with�a�Fitting�subgroup�of�class�3.���+�These�(Mf��gamilies�are�split�up�further�(Linto�subf�amilies�in�[Dek96]�and�to�each�subf�amily�is�assigned�(La�type;�that�is,�a�string���which��is��used�to�identify�the�subf��gamily��Y�.�As�mentioned�abo�٠v�e,�for��the�3-dimensional�almost�crystallographic�groups���the�� type�is�a�string�representing�the�numbers�from�1�to�17,�i.e.�the�a���v�� ailable�types�are��"01"�,��"02"�,���A:���:�:��$��,��"17"�.���F�٠or�!�the�4-dimensional�!�almost�crystallographic�groups�with�a�Fitting�subgroup�of�class�2�the�type�is�a�string�of�3�or���4�V4characters.�V3In�general,�a�string�of�3�characters�representing�the�number�of�the�table�entry�in�[Dek96]�is�used.�So���possible���types�are��"001"�,����"002"�,���A:���:�:��d/�.�The�reader�is�w��garned�ho�� we�v�٠er���that�not�all�possible�numbers�are�used,�e.g.���there�n�are�no�n�groups�of�type��"016"�.�Also,�the�types�do�not�appear�in�their�natural�order�in�[Dek96].�Moreo�٠v�er���,�n�for���certain�v~numbers�vthere�is�more�than�one�f��gamily�of�groups�listed�in�[Dek96].�F�٠or�e�xample,�v~the�3�f��gamilies�of�groups���corresponding���to�number�19�on���pages�179-180�of�[Dek96]�ha���v�٠e�types��"019"�,��"019b"��and��"019c"��(the�order�is�the���one�� gi�� v�٠en�in�[Dek96]).���ԍF�٠or���the�last�cate�gory�of�groups,�the�4-dimensional�almost���crystallographic�groups�with�a�Fitting�subgroup�of�class�3,���the���type�is�a�string�of�2�or�3�characters,�where�the�rst�character�is�al��gw�ays���the�letter��"B"�.�This��"B"��is�follo�� wed�by�the���number�� of�the�table�entry�as�found�in�[Dek96],�e�� v�٠entually�follo�wed�by�a��"b"��or��"c"��as�in�the�pre�vious�case.��� �� U̍�� ��10� 㮥�LChapter�� 3.�The�catalo��gg�of�almost�crystallo�gr�٠aphic�gr���oups���p ���� ��F�٠or���each���type�of�almost�crystallographic�group�contained�in�the�library�there�e�٠xists�a�function�taking�a�parameter�� �list�z�as�input�and�returning�the�desired�matrix�or�pcp�z�group.�These�functions�can�be�accessed�from��QGAP��using�the���lists���ACDim3Funcs�,���ACDim4Funcs�,��ACPcpDim3Funcs��and��ACPcpDim4Funcs��which�consist�of�the�corresponding���functions.��N8�Although��.we��-include�these�direct�access�functions�here�for�completeness,�we�note�that�the�user�should�in�general�use���the�-?higher���-le�� v�٠el�functions�introduced�abo�v�e�to�obtain�almost�crystallographic�groups�from�the�library��Y�.�In�particular���,���these�Q�lo�� w-le�v�٠el�Q�access�functions�return�matrix�or�pcp�groups,�b���ut�the�almost�crystallographic�info� ags�will�not�be���attached�� to�them.�� ��9��gap>�?�ACDim3Funcs[15];����9�function(�?�k1,�k2,�k3,�k4�)�...�end����9�gap>�?�ACDim3Funcs[15](1,1,1,1);����9�<matrix�?�group�with�5�generators>����9�gap>�?�ACPcpDim3Funcs[1](1);����9�Pcp-group�?�with�orders�[�0,�0,�0�]�� ��P3.4�� The��&electr��uonic�ver��sus�the�printed�librar�!Fy��N8��The�`package��aclib��can�_be�considered�as�the�electronic�v�٠ersion�of�Chapter�7�of�[Dek96].�In�this�section�we�outline���the��}relationship�between�the��|library�presented�in�this�manual�and�the�printed�v�٠ersion�in�[Dek96].�First�we�consider�an���e�٠xample.�� At�page�175�of�[Dek96],�we�nd�the�follo�� wing�groups�in�the�table�starting�with�entry�\13".���13.�� �LQ���=��LP�2�A=�Lc��: ���� ���m1?E���:��0�Dh�La�A;����Lb�A;��Lc�A;��Ld�E�A;�� �x;����Dj���� �D��[�Lb�A;����La�]��=�1���"aJ[�Ld�E�A;����La�]��=�1����b��Di���5 ����� ���� �D��[�Lc�A;����La�]��=��Ld��E��^��2�Tk����@�V zptmcm7t�@1����������[�Ld�E�A;����Lb�]��=�1�������� �D�[�Lc�A;����Lb�]��=�1�������[�Ld�E�A;����Lc�]��=�1������� �D��A� �x�La���=��La���^��E �1�� p]�A��Ld��E��^��Tk����@2��������A�� �x��^��2�����=���Ld��E��^��Tk����@3�������� �D��A� �x�Lb���=��Lb�A������ �x�Ld�0�=���Ld�E�A������ �D�� �x�Lc���=��Lc���^��E �1�� p]�A��Ld��E��^��E �2�Tk����@6�������� �D��A�:��La���=��La���^��E �1�� p]�A��Ld��E��^��Tk����@1�� �+�Tk����@2��������A��:���^��2����=���Ld��E��^��Tk����@5�������� �D��A�:��Lb���=��Lb���^��E �1�� p]�A��Ld��E��^��Tk����@4��������A�:��Ld�0�=���Ld�E�A������ �D��:��Lc���=��Lc���^��E �1�� p]�A��Ld��E��^��E �2�Tk����@6��������A� �x���=���Lc�A�:���Ld��E��^��Tk����@6������������=�8� � ��3���A�:��(�A� �x�)��=�����h���G�Cq zptmcm7v�G0������B�� ��B����B��cЍ�@����� ��� 1��1�������٤��d��Tk����@1���d�ٌ� f` ��� ��M��2�����&��+�8��Lk����2����C[A�0���R[A�D �2�Lk����6���������Ѝ�r�,�Tk����@3���r�,��`� f` ��� ��M��2�����|K��+������Ѝ�l�Tk����@6���l��`� f` ��� ��M��2���������� 1��0���$c��D �1���C[A0���Zv0���}�n0������ 1�0���(F�0���C[A1���Zv0���}�n0������ 1�0���(F�0���C[A0���V���D �1�������`��~nn�1��~nn��Љ f` � �� �2���������� 1��0���(F�0���C[A0���Zv0���}�n1��������h�� �v��G1������ �v�C�� �� �v�C���� �v�C��cЍ� �v�A���� �oc�A��(�A��)�=�����h���G0������B�� ��B����B��cЍ�@����� ��� 1��1���1��Lk����1��8��+�8��Lk����2����Et�Lk����4����Z��D �2�Lk����6���������Ѝ�zw��Tk����@5���zw۟�`� f` ��� ��M��2���������� 1��0���$[@�D �1���G,d0���b)�0���{�0������ 1�0���(=�0���CI��D �1���b)�0���{�0������ 1�0���(=�0���G,d0���^G@�D �1���{�0������ 1�0���(=�0���G,d0���b)�0���{�1��������h�� �qY�G1������ �qYC�� �� �qYC���� �qYC��cЍ� �qYA�����&�8��� ���LH��s0�����2��s0�(�LQ�A;����Z�)��=��Z�8��D��(�Z����2�� �)������4����=���Z������6���A=�LA�A;�� �� ����LA���=��Df�(�Lk����1�� �A;����:�:�:����;����Lk����6���)�Dj�Lk����1����=��0�A;�q��Lk����2���A;��:�:�:����;��Lk����5����D2���2�Z�A;�q��Lk����6���D2��Z�Dg����AB-groups:��N8��D8�Lk���A>���0�A;�q��Lk��D��0��mod��U�2�A;��q��(�Lk�+�A;����L0�A;��L1�A;��L0�A;��L1�A;��L0�)�����The� ,number� +\13"�at�the�be�٠ginning�of�this�entry�is�the�type�of�the�almost�crystallographic�group�in�this�library��Y�.�This�� �f��gamily��of��groups�with�type�13�depends�on�6�parameters��Lk����1�� �A;����Lk����2���A;��:�:�:����;��Lk����6����and��these��are�the��Lpar�٠ameter��gs��list�in�this�library��Y�.���The�Thrational�matrix�representation�in��QGAP��corresponds�e�٠xactly�to�the�printed�v�ersion�in�[Dek96]�where�it�is�named��A�:��.���In�� the�e�٠xample�belo�� w��Y�,�we�consider�the�group�with�parameters��(�Lk����1�� �A;����Lk����2���A;��Lk����3���A;��Lk����4���A;��Lk����5���A;��Lk����6���)��=�(8�A;��0�A;��1�A;��0�A;��1�A;��0)�.��� �� d���� ��LSection�� 4.�The�electr���onic�ver��gsus�the�printed�libr�٠ary� ���11���p ���� ��9��gap>�?�G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]);�� ��9�<matrix�?�group�with�6�generators>����9�gap>�?�G.5;����9�[�?�[�1,�4,�0,�0,�1/2�],�[�0,�-1,�0,�0,�0�],�[�0,�0,�1,�0,�0�],������[�?�0,�0,�0,�-1,�1/2�],�[�0,�0,�0,�0,�1�]�]����9�gap>�?�G.6;����9�[�?�[�1,�8,�0,�0,�1/2�],�[�0,�-1,�0,�0,�0�],�[�0,�0,�-1,�0,�0�],������[�?�0,�0,�0,�-1,�0�],�[�0,�0,�0,�0,�1�]�]��N8��F�٠or�BSa�4-dimensional�almost�crystallographic�BRgroup�the�matrix�group�is�b���uilt�up�such�that��Df�La�A;����Lb�A;��Lc�A;��Ld�E�A;�� �x;��:�;� ��Dg�BS�as�de-���scribed��8in�[Dek96]�forms�the�dening��9generating�set�of��LG�.�F�٠or�certain�types�the�elements��A� �x�,��A��or��A ����may�not�be���present.�� Similarly��Y�,�for�a�3-dimensional�group�we�ha���v�٠e�the�generating�set��Df�La�A;����Lb�A;��Lc�A;�� �x;��:��Dg�� �and��A�Tx�and��A����may�be�absent.��N8�T��37o�\�obtain�\�a�polyc�٠yclic�generating�sequence�from�the�dening�generators�of�the�matrix�group�we�ha���v�٠e�to�order�the���elements�Hin�the�Hgenerating�set�suitably��Y�.�F�٠or�this�purpose�we�tak��ge�the�subsequence�of��(�A � �x;����:�;��;��La�A;��Lb�A;��Lc�A;��Ld�E�)�H�of�Hthose�gener���-���ators���which���are�present�in�the�dening�generating�set�of�the�matrix�group.�This�ne�� w�ordering�of�the�generators�is�then���used�� to�dene�a�polyc�٠yclic�presentation�of�the�gi�� v�en�almost�crystallographic�group.��� �� w���� ��p ���YƢ�������M4����P ��?��NExample��, �� ��acomputations����8with��.almost���� ��!cr�\%ystallographic����Z�qgr��G�oups����>�ʍ�P4.1�� Example��&computations�I��N8��Using���the�functions�a���v�� ailable�for�pcp�groups�in�the�share�package����Qpolycyclic��it�is�no�w�easy�to�redo�some�of�the�� �calculations�w�of�w�[Dek96].�As�a�rst�e�٠xample�we�check�whether�the�groups�indicated�as�torsion�free�in�[Dek96]�are�also���determined���as�torsion���free�ones�by��QGAP�.�In�[Dek96]�these�almost�Bieberbach�groups�are�listed�as�\��37AB-groups".�So���for�J�type�J�\013"�these�are�the�groups�with�parameters��(�Lk�+�A;����0�A;��1�A;��0�A;��1�A;��0)�J��where��Lk�v�is�an�e�� v�٠en�inte�ger��s8.�J�Let'�s�look�J�at�some���e�٠xamples�� in��QGAP�:�� ��9��gap>�?�G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�true����9�gap>�?�G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�false����Further���,��there�is�also�some�cohomology�information�in�the�tables�of�[Dek96].�In�f��gact,�the�groups�in�this�library�were���obtained�� as�e�٠xtensions��LE����of�the�form��N8��� �Ɵ�1���D!��Z��D!��LE���D!��LQ��D!��1��N8��where,��in�the��4-dimensional�case��LQ����=��LE�:��A=�Dh�Ld�E�Di�.��The�cohomology�information�for�the�particular�e�٠xample�abo�v�e�sho�� ws���that��the��groups�determined�by�a�parameter�set��(�Lk����1�� �A;����Lk����2���A;��Lk����3���A;��Lk����4���A;��Lk����4���A;��Lk����6���)���are��equi�� v�alent�as��e�٠xtensions�to�the�groups�deter���-���mined�� by�the�parameters��(�Lk����1�� �A;����Lk����2�����mod��U�2�A;��Lk����3�����mod��2�A;��Lk����4�����mod��2�A;��Lk����5�����mod��2�A;��0)�.�� This�is�also�visible�in�nding�torsion:��� �� }���� ��LSection�� 3.�Example�computations�III�80��13���p ���� ��9��gap>�?�G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);�� ��9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�false����9�gap>�?�G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);����9�Pcp-group�?�with�orders�[�2,�2,�0,�0,�0,�0�]����9�gap>�?�IsTorsionFree(G);����9�true�� ��P4.2�� Example��&computations�II��N8��The�"�computation�"�of�cohomology�groups�played�an�important�role�in�the�classication�of�the�almost�Bieberbach�groups���in��[Dek96].��Using��QGAP�,�it�is�no�� w�possible�to�check�these�computations.�As�an�e�٠xample�we�consider�the�4-dimensional���almost��Rcrystallographic��Qgroups�of�type�85�on�page�202�of�[Dek96].�This�group��LE�10�has�6�generators.�In�the�table,�one���also�� nds�the�information��N8��� �]��LH��s0�����2��s0�(�LQ�A;����Z�)��=��Z�8��D��(�Z����2�� �)������2��8��D��Z����4���N8��for� ��LQ����=����LE�:��A=�Dh�Ld�E�Di� ��as�abo�٠v�e.� �Moreo�v�er���,�the��LQ�{module� ��Z��is�in�f��gact�the�group��Dh�Ld�E�Di�,�where�the��LQ�-action�comes�from���conjug��7ation��inside���LE�:��.�In�the�case�of�groups�of�type�85,��Z��is�a�tri�� vial��LQ�-module.�The�follo�� wing�e�٠xample�demonstrates���ho�� w�� to�(re)compute�this�tw��go-cohomology�group��LH��s0��^��2��s0�(�LQ�A;����Z�)�.�� ��9��gap>�?�G:=AlmostCrystallographicPcpGroup(4,�"085",�false);����9�Pcp�?�group�with�orders�[�2,�4,�0,�0,�0,�0�]����9�gap>�?�GroupGeneratedByd:=Subgroup(G,�[G.6]�);����9�Pcp�?�group�with�orders�[�0�]����9�gap>�?�Q:=G/GroupGeneratedByd;����9�Pcp�?�group�with�orders�[�2,�4,�0,�0,�0�]����9�gap>�?�action:=List(�Pcp(Q),�x�->�[[1]]�);����9�[�?�[�[�1�]�],�[�[�1�]�],�[�[�1�]�],�[�[�1�]�],�[�[�1�]�]�]����9�gap>�?�C:=CRRecordByMats(�Q,�action);;����9�gap>�?�TwoCohomologyCR(�C�).factor.rels;����9�[�?�2,�2,�4,�0�]����This�M�last�line�gi�� v�٠es�M�us�the�abelian�in���v�ariants�of�the�second�M�cohomology�group��LH��s0��^��2��s0�(�LQ�A;����Z�)�.�So�we�should�read�this�line�as��N8��� ��c�LH��s0�����2��s0�(�LQ�A;����Z�)��=��Z����2��8��D�8��Z����2���D��Z����4���D��Z��N8��which�� indeed�coincides�with�the�information�in�[Dek96].��N8��P4.3�� Example��&computations�III����As���another�application���of�the�capabilities�of�the�combination�of��aclib��and��Qpolycyclic��we�check�some�computations���of�� [DM01].��N8�Section�ay5�of�azthe�paper�[DM01]�is�completely�de�� v���oted�to�an�e�٠xample�of�the�computation�of�the��LP�-localization�of�a���virtually��Cnilpotent�group,�where��D�LP��is�a�set�of�primes.�Although�it�is�not�our�intention�to�de�� v�٠elop�the�theory�of��LP�-���localization�� of�groups�at�this�place,�let�us�summarize�some�of�the�main�results�concerning�this�topic�here.���F�٠or���a���set�of�primes��LP�,�we�say�that��Ln�JD�D2�JE�LP��if�and�only�if��Ln��is�a�product�of�primes�in��LP�.�A���group��LG��is�said�to�be��LP�-local���if�Xand�Yonly�if�the�map��A����Tn�����:����LG��D!��LG��:��Lg��D7!��Lg���^��Tn��Y�is�Xbijecti�� v�٠e�for�all��Ln����D2��LP���^��E0��l��,�where�X�LP���^��E0���$�is�the�set�of�all�primes�not�in��LP�.���The�Z#�LP�-localization�Z"of�a�group��LG�,�is�a��LP�-local�group��LG����TP��!�together�with�a�morphism��A�/��:�[�LG�[ �D!��LG����TP��!�which�satisfy�Z#the��� �� ����� ��14� �@��LChapter�� 4.�Example�computations�with�almost�crystallo��ggr�٠aphic�gr���oups���p ���� ��follo�� wing�FKuni�v�٠ersal�FLproperty:�F�or�FLeach��LP�-local�group��LL�M��and�an�٠y�morphism��A'����:���LG��D!��LL���,�there�FLe�٠xists�a�unique�morphism�� ��A ����:���LG����TP����D!��LL���,�� such�that��A � X�D�8��A��=���A'� �x�.����This�Âconcept�of�Ãlocalization�is�well�de�� v�٠eloped�for�nite�groups�and�for�nilpotent�groups.�F�٠or�a�nite�group��LG�,�the��LP�-���localization��is��the�lar���gest�quotient�of��LG�,�ha���ving�no�elements�with�an�order�belonging�to��LP���^��E0���V�(the�morphism��A� �x�,�mentioned���abo�٠v�e�� is�the�natural�projection).����In���[DM01]�a�contrib���ution�is�made�to�� w��gards�the���localization�of�virtually�nilpotent�groups.�The�theory�de�v�٠eloped�in���the��gpaper�is�then�illustrated�in�the�last�section�of�the��fpaper�by�means�of�the�computation�of�the��LP�-localization�of���an��almost�crystallographic�group.��F�٠or�their�e�xample�the�authors�ha���v�e�chosen�an��almost�crystallographic�group��LG����of���dimension���3�and�type�17.�F�٠or�the�set�of�parameters��(�Lk����1�� �A;����Lk����2���A;��Lk����3���A;��Lk����4���)����the�٠y���ha���v�e�considered���all�cases�of�the�form����(�Lk����1�� �A;����Lk����2���A;��Lk����3���A;��Lk����4���)��=�(2�A;��0�A;��0�A;��Lk����4�� �)�.���Here���we���will�check�their�computations�in�tw��go�cases��Lk����4�� ��=��0��and��Lk����4���=��1��using���the�set�of�primes��LP���=��Df�2�Dg�.���The���holonomy�(group�of�these�almost�crystallographic�groups��LG��is�the�dihedral�)group��DD����6��(�of�order�12.�Thus�there�is�a�short���e�٠xact�� sequence�of�the�form���� ����1���D!���Fitt���e(�LG�)��D!��LG��D!�D����6����D!��1�D��z ��As��a��rst�step�in�their�computation,�Descheemaek��ger�and�Malf��gait�determine�the�group��LI����TP����"�Fƛ� zptmcm7y�F0������DD����6�� �,�which�is�the�unique���subgroup��yof�order�3�in��DD����6�� �.�One�of��xthe�main�objects�in�[DM01]�is�the�group��LK���=����Lp���^��E �1�� p]�(�LI����TP����"�F0������DD����6���)�,�where��Lp��is�the�natural���projection��of���LG��onto�its�holonomy�group.�It�is�kno�� wn�that�the��LP�-localization�of��LG��coincides�with�the��LP�-localization���of���LG�A= ����3�� �(�LK� ���)�,�where��A ����3���(�LK� ���)��is�the�third�term�in�the�lo�� wer�central�series�of��LK��.�As��LG�A= ����3�� �(�LK��)��is�nite�in�this�e�٠xample,�we���e�٠xactly�kno�� w�what�this��LP�-localization�is.�Let�us�no�� w�sho�w��Y�,�ho�w�GAP� �can�be�used�to�compute�this��LP�-localization�in���tw��go�� cases:����First�� case:�The�parameters�are��(�Lk����1�� �A;����Lk����2���A;��Lk����3���A;��Lk����4���)��=�(2�A;��0�A;��0�A;��0)�����9��gap>�?�G�:=�AlmostCrystallographicPcpGroup(3,�17,�[2,0,0,0]�);����9�Pcp�?�group�with�orders�[�2,�6,�0,�0,�0�]����9�gap>�?�projection�:=�NaturalHomomorphismOnHolonomyGroup(�G�);����9�[�?�g1,�g2,�g3,�g4,�g5�]�->�[�g1,�g2,�identity,�identity,�identity�]����9�gap>�?�F�:=�HolonomyGroup(�G�);����9�Pcp�?�group�with�orders�[�2,�6�]����9�gap>�?�IPprimeD6�:=�Subgroup(�F�,�[F.2^2]�);����9�Pcp�?�group�with�orders�[�3�]����9�gap>�?�K�:=�PreImage(�projection,�IPprimeD6�);����9�Pcp�?�group�with�orders�[�3,�0,�0,�0�]����9�gap>�?�PrintPcpPresentation(�K�);����9�pcp�?�presentation�on�generators�[�g2^2,�g3,�g4,�g5�]����9�g2^2�?�^�3�=�identity����9�g3�?�^�g2^2�=�g3^-1*g4^-1����9�g3�?�^�g2^2^-1�=�g4*g5^-2����9�g4�?�^�g2^2�=�g3*g5^2����9�g4�?�^�g2^2^-1�=�g3^-1*g4^-1*g5^2����9�g4�?�^�g3�=�g4*g5^2����9�g4�?�^�g3^-1�=�g4*g5^-2����9�gap>�?�Gamma3K�:=�CommutatorSubgroup(�K,�CommutatorSubgroup(�K,�K�));����9�Pcp�?�group�with�orders�[�0,�0,�0�]����9�gap>�?�quotient�:=�G/Gamma3K;����9�Pcp�?�group�with�orders�[�2,�6,�3,�3,�2�]����9�gap>�?�S�:=�SylowSubgroup(�quotient,�3);����9�Pcp�?�group�with�orders�[�3,�3,�3�]����9�gap>�?�N�:=�NormalClosure(�quotient,�S);����9�Pcp�?�group�with�orders�[�3,�3,�3�]����9�gap>�?�localization�:=�quotient/N;��� �� �썟� ��LSection�� 3.�Example�computations�III�80��15���p ���� ��9��Pcp�?�group�with�orders�[�2,�2,�2�]�� ��9�gap>�?�PrintPcpPresentation(�localization�);����9�pcp�?�presentation�on�generators�[�g1,�g2,�g3�]����9�g1�?�^�2�=�identity����9�g2�?�^�2�=�identity����9�g3�?�^�2�=�identity�� ��This�� sho�� ws�that��LG����TP�����W�����D���(���=�����1�Z���^���3��d�2��� �.��N8�Second�� case:�The�parameters�are��(�Lk����1�� �A;����Lk����2���A;��Lk����3���A;��Lk����4���)��=�(2�A;��0�A;��0�A;��1)����9��gap>�?�G�:=�AlmostCrystallographicPcpGroup(3,�17,�[2,0,0,1]);;�� ��9�gap>�?�projection�:=�NaturalHomomorphismOnHolonomyGroup(�G�);;����9�gap>�?�F�:=�HolonomyGroup(�G�);;����9�gap>�?�IPprimeD6�:=�Subgroup(�F�,�[F.2^2]�);;����9�gap>�?�K�:=�PreImage(�projection,�IPprimeD6�);;����9�gap>�?�Gamma3K�:=�CommutatorSubgroup(�K,�CommutatorSubgroup(�K,�K�));;����9�gap>�?�quotient�:=�G/Gamma3K;;����9�gap>�?�S�:=�SylowSubgroup(�quotient,�3);;����9�gap>�?�N�:=�NormalClosure(�quotient,�S);;����9�gap>�?�localization�:=�quotient/N;����9�Pcp�?�group�with�orders�[�2,�2,�2�]����9�gap>�?�PrintPcpPresentation(�localization�);����9�pcp�?�presentation�on�generators�[�g1,�g2,�g3�]����9�g1�?�^�2�=�identity����9�g2�?�^�2�=�g3����9�g3�?�^�2�=�identity����9�g2�?�^�g1�=�g2*g3����9�g2�?�^�g1^-1�=�g2*g3����In�� this�case,�we�see�that��LG����TP����=���DD����4�� �.���The��reader�can��check�that�these�results�coincide�with�those�obtained�in�[DM01].�Note�also�that�we�used�a�some�� what���dierent�scheme�to�compute�this�localization�than�the�one�used�in�[DM01].�W��37e�in���vite�the�reader�to�check�the�same���computations,�� tracing�e�٠xactly�the�steps�made�in�[DM01].��� �� �Y��� ��p ����C���� ���NBib���liograph��G�y����<!�����:��[Aus60]��$ Louis��CAuslander��s8.��BBieberbach'�s�theorem��Bon�space�groups�and�discrete�uniform�subgroups�of�lie�groups.��KAnn.�� ��$ of�� Math�,�71(3):579{590,�1960.�������W[Bro82]��$ K�� enneth��*S.�Bro�wn.��KCohomology��)of�Groups�,�v���olume�87�of��KGrad.�T��L�e�٠xts�in�Math.��Springer���-V����erlag,�Ne�� w�Y���gork,����$ 1982.��������[DE01a]��$ Karel��oDekimpe��nand�Bettina�Eick.�Computational�aspects�of�group�e�٠xtensions�and�their�application�in����$ topology��Y�.�� �KSubmitted�,�2001.�������[DE01b]��$ Karel�� Dekimpe�and�Bettina�Eick.�Computations�with�almost�crystallographic�groups.��KSubmitted�,�2001.������� [Dek96]��$ Karel��jDekimpe.��k�KAlmost-Bieberbach�Groups:�Ane�and�Polynomial�Structures�,�v���olume�1639�of��KLecture����$ 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