In my thesis I shall be investigating two distinct metaplectic extensions of the general linear group. The first of these was discovered by Matsumoto, it's existence intimately connected with the deep properties of the r-th order Hilbert symbol. His construction relies heavily on class field theory and algebraic K-theory. Having constructed his metaplectic group, which is known to be universal, Matsumoto was then able to define the cocycle representing this extension. The second of these metaplectic extensions was found recently by Dr Hill at University College London. In contrast, his construction is very elementary. He was able to prove the existence of a continuous cocycle resulting in the construction of a new non-trivial metaplectic extension. It has already been shown, by Hill, that these two metaplectic extensions are isomorphic if we restrict to the special linear group. However, little is known of this isomorphism. Throughout this thesis we shall investigate these two cocycles, finding explicit formulae in both cases. We shall then show that the isomorphism between the group extensions of Matsumoto and Hill may be defined via the discovery of the coboundary which splits the quotient of the corresponding cocycles. Having found this coboundary we shall then be able to prove that, in specific cases, the two extensions are in fact isomorphic over the full general linear group.

No abstract / Mathematical Sciences / (D. Phil. (Mathematics))

No abstract / Mathematical Sciences / (D. Phil. (Mathematics))

Ce travail est consacré à l'étude de systèmes de Lyndon (pour la relation d'ordre alterné) et àla dynamique symbolique des développements des nombres en base négative. Pour un réel ß > 1fixé, nous construisons un code préfixe récurrent positif permettant non seulement de montrerl'intrinsèque ergodicité du —ß-shift mais aussi de déterminer la fonction zêta qui lui est associée.Nous étudions les conditions pour lesquelles le —ß-shift possède la spécification.En outre, lorsque ß est strictement plus petit que le nombre d'or, le langage du —ß-shift admet desmots intransitifs. Cet état de fait engendre dans le système dynamique des cylindres négligeablespar rapport à la mesure d'entropie maximale. Ces cylindres génèrent sur Iß=[—ß/(ß+1),1/(ß+1)[ depetits intervalles de mesure nulle (la mesure considérée étant l'unique mesure ergodique sur Iß).Nous en faisons une étude détaillée, en particulier nous déterminons ces intervalles "trous".Par ailleurs, nous étudions l'unicité des systèmes de numération des entiers relatifs en base négative et nous montrons qu'à chaque mot de Lyndon correspond un tel système. / This work deals with the study of the Lyndon systems (for alternate order) and the symbolicdynamics of the expansions of real numbers in negative base. For a given real ß > 1, we showthe intrinsic ergodicity of the —ß-shift using a positive recurring prefix code and we determine theassociated zeta function. We study the conditions for which the —ß-shift admits the specificationproperty.Moreover, when ß is less than golden ratio, the language of the —ß-shift contains intransitive words.These words lead to some cylinders negligible with respect to the measure with maximal entropy.In the interval Iß=[—ß/(ß+1),1/(ß+1)[, these cylinders correspond to some gaps: small interval withmeasure zero (with respect to the unique ergodic measure on Iß). We make a detailed study ofthese gaps.Otherwise, we study the uniqueness of the number systems of integers in negative base and weshow that to each Lyndon word corresponds to a such system.