By Boris N. Apanasov
This ebook offers a scientific account of conformal geometry of n-manifolds, in addition to its Riemannian opposite numbers. A unifying subject is their discrete holonomy teams. particularly, hyperbolic manifolds, in size three and better, are addressed. The therapy covers additionally proper topology, algebra (including combinatorial staff concept and forms of staff representations), mathematics matters, and dynamics. growth in those parts has been very quick sicne the Eighties, specifically end result of the Thurston geometrization software, resulting in the answer of many tricky difficulties. a powerful attempt has been made to indicate new connections and views within the box and to demonstrate quite a few features of the idea. An intuitive technique which emphasizes the information in the back of the structures is complemented through a good number of examples and figures which either use and help the reader's geometric mind's eye.
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Extra info for Conformal Geometry of Discrete Groups and Manifolds (Degruyter Expositions in Mathematics)
4. Lie subgroups of the Mobius group 19 Now, as for the Gieseking manifold, we realize those 3-simplices as regular ideal simplices in the hyperbolic 3-space H3 and consider orientation preserving isometries of l l which identify sides of those simplices as indicated in Figure 8. , a hyperbolic manifold) and homeomorphic to S3\K. One can observe that this non-compact (finite volume) oriented hyperbolic 3manifold is the two-fold covering of the (non-oriented) Gieseking manifold. In other sections, we will discuss another hyperbolic structures, in particular, on the Borromean and Whitehead link complements in the 3-sphere S3.
These groups were classified by Goursat  and H. Hopf . For a very careful and accessible description of all 3-dimensional spherical forms, we refer a reader to Postnikov . 6. The eight 3-dimensional geometries 27 manifolds one can find RIP3, the Poincare homology sphere, prism manifolds and lens spaces L(p, q); all of them are Seifert fibered spaces, see Scott . G-2. Euclidean (= flat) geometry. As any isometry g E Isom R3 can be expressed as g(x) = U(x) + b, U E 0(3) and x, b E R3, the map g H U defines a surjective homomorphism Isom 1R3 -+ 0(3) with the group of translations of R3 as the kernel.
This is related to the theory of quasiconformal mappings in space, see Vaisala  and Reshetnyak , while the compact- ness principle is originally due to Belinskii [1, 2]. 5 is due to GehringMartin ; it reformulates the compactness principle for quasiconformal mappings in terms of convergence groups, see Chapter 2. For the linear representation of the Mobius group we follow Apanasov ; see also Kobayashi-Nomizu  and Greenberg . 12 with a classification of Lie subgroups of the Mobius group Mob(n).