Thesis (Ph.D.) - University of St Andrews, November 2007.

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In this thesis we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups. This correspondence between Q-powered nilpotent groups and rational nilpotent Lie algebras was discovered by Anatoly Mal'cev in 1951. We show how the Mal'cev correspondence can be realized on a computer. We explore two possibilities for this purpose and compare them: the first one uses matrix embeddings and the second the Baker-Campbell-Hausdorff formula. Then, we describe a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection. Algorithms for collection lie at the heart of most methods dealing with polycyclically presented groups. The current state of the art is "collection from the left" as recently studied by Gebhardt, Leedham-Green/Soicher and Vaughan-Lee. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory. Further, we explore how the Mal'cev correspondence can be used to describe symbolically the collection process in polycyclically presented groups. In particular, we describe an algorithm that computes the collection functions for splittable polycyclic groups. This algorithm is based on work by du Sautoy. We apply it to the computation of pro-p-completions of polycyclic groups. Finally we describe a practical algorithm for testing polycyclicity of finitely generated rational matrix groups. Previously, not only did no such method exist but it was not clear whether this question was decidable at all. Most of the methods described in this thesis are implemented in the computer algebra system GAP and publicly available as part of the GAP packages Guarana and Polenta. Reports on the implementation including runtimes for some examples are given at the appropriate places.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-05-18T18:27:17Z No. of bitstreams: 2 Dissertação - Jefferson dos Santos e Silva - 2015.pdf: 741852 bytes, checksum: 8cb431ec9a186100784d60268a133fcf (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-05-18T18:28:34Z (GMT) No. of bitstreams: 2 Dissertação - Jefferson dos Santos e Silva - 2015.pdf: 741852 bytes, checksum: 8cb431ec9a186100784d60268a133fcf (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-18T18:28:34Z (GMT). No. of bitstreams: 2 Dissertação - Jefferson dos Santos e Silva - 2015.pdf: 741852 bytes, checksum: 8cb431ec9a186100784d60268a133fcf (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-19 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work, based on [9], describes an effective method for computing a consistent polycyclic presentation for the nonanbeian tensor square G G of a group G given by a consistent polycyclic presentation. / Este trabalho, baseado em [9], determina um efetivo método para calcular uma apresentação policíclica consistente para o quadrado tensorial não abeliano G G de um grupo G dado por uma apresentação policíclica consistente.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T15:51:20Z No. of bitstreams: 2 Dissertação - Laredo Rennan Pereira Santos - 2015.pdf: 970933 bytes, checksum: 6b8836c42db993ababe18805a5857373 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T15:54:12Z (GMT) No. of bitstreams: 2 Dissertação - Laredo Rennan Pereira Santos - 2015.pdf: 970933 bytes, checksum: 6b8836c42db993ababe18805a5857373 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-05-19T15:54:12Z (GMT). No. of bitstreams: 2 Dissertação - Laredo Rennan Pereira Santos - 2015.pdf: 970933 bytes, checksum: 6b8836c42db993ababe18805a5857373 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we consider monoid presentations MonhX,Ri, with set of generators X and relations R, defining groups and monoids as equivalence classes of words over X, in relation to a congruence generated by R. Taking R as a rewriting system with respect to an linear ordering of X , the set of words over X, we can apply some rewriting strategies in its laws. We use a version of the Knuth-Bendix method in R to find a confluent rewriting system equivalent to original, when such finite system exist. This new set of relations, denoted by RC(X, R), allows that in MonhX,Ri any element be defined by a unique irreducible word with respect to RC(X, R). We exhibit several examples of the execution of the Knuth-Bendix method from the functions of KBMAG package of the GAP system. Lastly, we set up a sufficient condition so that certain monoid presentations for polycyclic groups be confluent. / Neste trabalho, consideramos apresentações monoidais MonhX,Ri, com conjunto de geradores X e de relações R, definindo grupos e monoides como classes de equivalência de palavras sobre X, em relação a uma congruência gerada por R. Tomando R como um sistema de reescrita com respeito à uma ordenação linear de X , o conjunto de palavras sobre X, podemos aplicar algumas estratégias de reescrita em suas leis. Usamos uma versão do método de Knuth-Bendix em R para encontrar um sistema de reescrita confluente que seja equivalente ao original, quando um tal sistema finito existe. Este novo conjunto de relações, denotado por RC(X, R), permite que em MonhX,Ri qualquer elemento seja definido por uma única palavra irredutível com respeito a RC(X, R). Exibimos diversos exemplos da execução do método de Knuth-Bendix a partir das funções do pacote KBMAG do sistema GAP. Por fim, estabelecemos uma condição suficiente para que certas apresentações monoidais para grupos policíclicos sejam confluentes.