Browsing by Subject "Group theory"

Now showing 1 - 4 of 4

Higher-rank generalizations of convex cocompact and geometrically finite dynamics
(2022-06-29) Weisman, Theodore Joseph; Danciger, Jeffrey; Allcock, Daniel; Bowen, Lewis; Ballas, Samuel
We study several higher-rank generalizations of the dynamical behavior of convex cocompact groups in rank-one Lie groups, in the context of both convex projective geometry and relatively hyperbolic groups. Our results include a dynamical characterization of a notion of convex cocompact projective structure due to Danciger-Guéritaud-Kassel. This generalizes a dynamical characterization of Anosov representations of hyperbolic groups. Using topological dynamics, we also define a new notion of geometrical finiteness in higher rank which generalizes previous notions of relative Anosov representation due to Kapovich-Leeb and Zhu. We prove that these “extended geometrically finite” representations are stable under certain small relative deformations, and we provide various examples coming from the theory of convex projective structures.
The Neʼeman-Fairlie SU(2/1) model: from superconnection to noncommutative geometry
(2004) Asakawa, Takeshi; Fischler, Willy
The gauge symmetries provide the Standard Model, and it involves a Spontaneous Symmetry Breakdown (SSB), which is called the Higgs Mechanism. However, the geometrical formulation that can include the SSB and the unification of gauge groups have not solved yet. In order to solve these problems, we pay attention to two geometrical tools, that is, the superconnection and Noncommutative Geometry (NCG). And we consider the supergroup SU(2/1) as a candidate of the electroweak unification group. The SU(2/1) unification model was formulated with several geometrical frameworks, mainly, superconnection and the Mainz-Marseille (MM) version of NCG. On the other hand, the Connes-Lott (CL) model with an ‘original’ NCG was suggested without the supergroup SU(2/1). In these models the Higgs scalar field can be geometrically formulated. In this dissertation we attempt the new Ne’emanFairlie SU(2/1) model in the geometrical framework of the CL model. By assuming the 3 × 3 lepton and the 4 × 4 quark representations of SU(2/1) in the generalized on
On some residual and locally virtual properties of groups
(2010-05) Katerman, Eric Michael; Reid, Alan W.; Gordon, Cameron; Luecke, John; Allcock, Daniel; G�l, Anna
We define a strong form of subgroup separability, which we call RS separability, and we use this to combine LERF and Agol’s RFRS condition on groups into a property called LVRSS. We show that some infinite classes of groups that are known to be both subgroup separable and virtually RFRS are also LVRSS. We also provide evidence for the naturalness of RS separability and LVRSS by showing that they are preserved under various operations on groups.
Regular realizations of p-groups
(2008-05) Hammond, John Lockwood; Saltman, D. J. (David J.), 1951-
This thesis is concerned with the Regular Inverse Galois Problem for p-groups over fields of characteristic unequal to p. Building upon results of Saltman, Dentzer characterized a class of finite groups that are automatically realized over every field, and proceeded to show that every group of order dividing p⁴ belongs to this class. We extend this result to include groups of order p⁵, provided that the base field k contains the p³-th roots of unity. The proof involves reducing to certain Brauer embedding problems defined over the rational function field k(x). Through explicit computation, we describe the cohomological obstructions to these embedding problems. Then by applying results about the Brauer group of a Dedekind domain, we show that they all possess solutions.