Normal subgroup

Abstract: 1. Statement of Main Results.- 2. Synopsis of the Chapters.- 3. Remarks on the Structure of the Book, References and Notation.- 1. Preliminaries.- 0. Notation, Terminology and Some Basic Facts.- 1. Algebraic Groups Over Arbitrary Fields.- 2. Algebraic Groups Over Local Fields.- 3. Arithmetic Groups.- 4. Measure Theory and Ergodic Theory.- 5. Unitary Representations and Amenable Groups.- II. Density and Ergodicity Theorems.- 1. Iterations of Linear Transformations.- 2. Density Theorems for Subgroups with Property (S)I.- 3. The Generalized Mautner Lemma and the Lebesgue Spectrum.- 4. Density Theorems for Subgroups with Property (S)II.- 5. Non-Discrete Closed Subgroups of Finite Covolume.- 6. Density of Projections and the Strong Approximation Theorem.- 7. Ergodicity of Actions on Quotient Spaces.- III. Property (T).- 1. Representations Which Are Isolated from the Trivial One-Dimensional Representation.- 2. Property (T) and Some of Its Consequences. Relationship Between Property (T) for Groups and for Their Subgroups.- 3. Property (T) and Decompositions of Groups into Amalgams.- 4. Property (R).- 5. Semisimple Groups with Property (T).- 6. Relationship Between the Structure of Closed Subgroups and Property (T) of Normal Subgroups.- IV. Factor Groups of Discrete Subgroups.- 1. b-metrics, Vitali's Covering Theorem and the Density Point Theorem.- 2. Invariant Algebras of Measurable Sets.- 3. Amenable Factor Groups of Lattices Lying in Direct Products.- 4. Finiteness of Factor Groups of Discrete Subgroups.- V. Characteristic Maps.- 1. Auxiliary Assertions.- 2. The Multiplicative Ergodic Theorem.- 3. Definition and Fundamental Properties of Characteristic Maps.- 4. Effective Pairs.- 5. Essential Pairs.- VI. Discrete Subgroups and Boundary Theory.- 1. Proximal G-Spaces and Boundaries.- 2. ?-Boundaries.- 3. Projective G-Spaces.- 4. Equivariant Measurable Maps to Algebraic Varieties.- VII. Rigidity.- 1. Auxiliary Assertions.- 2. Cocycles on G-Spaces.- 3. Finite-Dimensional Invariant Subspaces.- 4. Equivariant Measurable Maps and Continuous Extensions of Representations.- 5. Superrigidity (Continuous Extensions of Homomorphisms of Discrete Subgroups to Algebraic Groups Over Local Fields).- 6. Homomorphisms of Discrete Subgroups to Algebraic Groups Over Arbitrary Fields.- 7. Strong Rigidity (Continuous Extensions of Isomorphisms of Discrete Subgroups).- 8. Rigidity of Ergodic Actions of Semisimple Groups.- VIII. Normal Subgroups and "Abstract" Homomorphisms of Semisimple Algebraic Groups Over Global Fields.- 1. Some Properties of Fundamental Domains for S-Arithmetic Subgroups.- 2. Finiteness of Factor Groups of S-Arithmetic Subgroups.- 3. Homomorphisms of S-Arithmetic Subgroups to Algebraic Groups.- IX. Arithmeticity.- 1. Statement of the Arithmeticity Theorems.- 2. Proof of the Arithmeticity Theorems.- 3. Finite Generation of Lattices.- 4. Consequences of the Arithmeticity Theorems I.- 5. Consequences of the Arithmeticity Theorems II.- 6. Arithmeticity, Volume of Quotient Spaces, Finiteness of Factor Groups, and Superrigidity of Lattices in Semisimple Lie Groups.- 7. Applications to the Theory of Symmetric Spaces and Theory of Complex Manifolds.- Appendices.- A. Proof of the Multiplicative Ergodic Theorem.- B. Free Discrete Subgroups of Linear Groups.- C. Examples of Non-Arithmetic Lattices.- Historical and Bibliographical Notes.- References.

Abstract: It is pointed out that the definition of the inhomogeneous Lorentz group as a symmetry group breaks down in the presence of gravitational fields even when the dynamical effects of gravitational forces are completely negligible. An attempt is made to rederive the Lorentz group as an "asymptotic symmetry group" which leaves invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. By analyzing recent work of Bondi and others on gravitational radiation it is shown that, with apparently reasonable boundary conditions, one obtains not the Lorentz group but a larger group. The name "generalized Bondi-Metzner group" ("GBM group") is suggested for this larger group.It is shown that the GBM group contains an Abelian normal subgroup whose factor group is isomorphic to the homogeneous orthochronous Lorentz group; that the GBM group contains precisely one Abelian four-dimensional normal subgroup, which can be identified with the group of rigid translations; that the GBM group contains an infinite number of different subgroups isomorphic to the inhomogeneous orthochronous Lorentz group; that the infinitesimal GBM group algebra permits at least one nontrivial representation, which is directly analogous to the rest-mass-zero and spin-zero representation of the Lorentz group; that in any representation of the infinitesimal GBM group algebra there is a "rest mass" operator which commutes with all the other operations; and that no similar "spin" operator appears to exist. It is argued that the GBM group is so similar to the inhomogeneous Lorentz group that the former may be compatible as a symmetry group with present microphysics.Two applications are given: Certain quantum commutation relations covariant under GBM transformations are presented; and a denumerably infinite set of integral invariants, for classical asymptotically flat gravitational fields, are derived. The four simplest integral invariants constitute the total energy momentum radiated to infinity by gravitational waves.

Abstract: In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).