ArtinTits groups have fruitfully been studied since the thirties when Emil Artin introduced the family of braid groups in the context of knot theory. The precise definition of the whole family of ArtinTits groups is due to Jacques Tits in the seventies who introduced them as extensions of Coxeter groups. Although their definition is purely algebraic and relies on certain combinatorial data encoded in a so called "Coxeter graph", their study is not limited to their own field, but connections with other branch of mathematics has been discovered such as geometric group theory and algebraic topology. Since the definition of ArtinTits groups is given by means of Coxeter graphs, it's quite natural to wonder whether there exists any connection between the structure of the combinatorial data that is written in the graph and the algebraic properties of the associated group. Not surprisingly, this is sometimes the case and a number of results we present in this thesis state hypotheses on the structure of the Coxeter graph in order to infer algebraic information regarding the ArtinTits group associated to it. However, the "slightest variation" in the structure of the graph may change a lot the properties of the related ArtinTits group. This situation has the consequence that many general questions about these groups have been answered only for some more or less extended families using tools specifically developed for that cases. E.g., we do not know if all ArtinTits groups are torsion free and proofs are known only for some of them. In particular, in our thesis we want to study polyfreeness, a strong structural property of groups that in turn implies an affirmative answer to questions like torsionfreeness, orderability and others. We achieve that the only polyfree ArtinTits groups of finite type are those of type I_2(n) (n >= 3), A_3, B_3, B_4 and D_4 (with the only possible exception of F_4 which remains unknown at the moment). Moreover, we prove that all ArtinTits groups whose associated graph is a tree or a forest are polyfree and in the last part we give an affirmative result about polyfreeness of ArtinTits groups built starting from graphs that may be "more connected" than just a tree or a forest and that we hope to be able to further generalise in the future.
ArtinTits groups have fruitfully been studied since the thirties when Emil Artin introduced the family of braid groups in the context of knot theory. The precise definition of the whole family of ArtinTits groups is due to Jacques Tits in the seventies who introduced them as extensions of Coxeter groups. Although their definition is purely algebraic and relies on certain combinatorial data encoded in a so called "Coxeter graph", their study is not limited to their own field, but connections with other branch of mathematics has been discovered such as geometric group theory and algebraic topology. Since the definition of ArtinTits groups is given by means of Coxeter graphs, it's quite natural to wonder whether there exists any connection between the structure of the combinatorial data that is written in the graph and the algebraic properties of the associated group. Not surprisingly, this is sometimes the case and a number of results we present in this thesis state hypotheses on the structure of the Coxeter graph in order to infer algebraic information regarding the ArtinTits group associated to it. However, the "slightest variation" in the structure of the graph may change a lot the properties of the related ArtinTits group. This situation has the consequence that many general questions about these groups have been answered only for some more or less extended families using tools specifically developed for that cases. E.g., we do not know if all ArtinTits groups are torsion free and proofs are known only for some of them. In particular, in our thesis we want to study polyfreeness, a strong structural property of groups that in turn implies an affirmative answer to questions like torsionfreeness, orderability and others. We achieve that the only polyfree ArtinTits groups of finite type are those of type I_2(n) (n >= 3), A_3, B_3, B_4 and D_4 (with the only possible exception of F_4 which remains unknown at the moment). Moreover, we prove that all ArtinTits groups whose associated graph is a tree or a forest are polyfree and in the last part we give an affirmative result about polyfreeness of ArtinTits groups built starting from graphs that may be "more connected" than just a tree or a forest and that we hope to be able to further generalise in the future.
Polyfree constructions for some Artin groups
MORESCHINI, LORENZO
20191213
Abstract
ArtinTits groups have fruitfully been studied since the thirties when Emil Artin introduced the family of braid groups in the context of knot theory. The precise definition of the whole family of ArtinTits groups is due to Jacques Tits in the seventies who introduced them as extensions of Coxeter groups. Although their definition is purely algebraic and relies on certain combinatorial data encoded in a so called "Coxeter graph", their study is not limited to their own field, but connections with other branch of mathematics has been discovered such as geometric group theory and algebraic topology. Since the definition of ArtinTits groups is given by means of Coxeter graphs, it's quite natural to wonder whether there exists any connection between the structure of the combinatorial data that is written in the graph and the algebraic properties of the associated group. Not surprisingly, this is sometimes the case and a number of results we present in this thesis state hypotheses on the structure of the Coxeter graph in order to infer algebraic information regarding the ArtinTits group associated to it. However, the "slightest variation" in the structure of the graph may change a lot the properties of the related ArtinTits group. This situation has the consequence that many general questions about these groups have been answered only for some more or less extended families using tools specifically developed for that cases. E.g., we do not know if all ArtinTits groups are torsion free and proofs are known only for some of them. In particular, in our thesis we want to study polyfreeness, a strong structural property of groups that in turn implies an affirmative answer to questions like torsionfreeness, orderability and others. We achieve that the only polyfree ArtinTits groups of finite type are those of type I_2(n) (n >= 3), A_3, B_3, B_4 and D_4 (with the only possible exception of F_4 which remains unknown at the moment). Moreover, we prove that all ArtinTits groups whose associated graph is a tree or a forest are polyfree and in the last part we give an affirmative result about polyfreeness of ArtinTits groups built starting from graphs that may be "more connected" than just a tree or a forest and that we hope to be able to further generalise in the future.File  Dimensione  Formato  

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