By Boris N. Apanasov

ISBN-10: 3110144042

ISBN-13: 9783110144048

This e-book offers a scientific account of conformal geometry of n-manifolds, in addition to its Riemannian opposite numbers. A unifying subject matter is their discrete holonomy teams. specifically, hyperbolic manifolds, in size three and better, are addressed. The remedy covers additionally correct topology, algebra (including combinatorial staff idea and different types of crew representations), mathematics matters, and dynamics. growth in those components has been very quick sicne the Nineteen Eighties, specifically because of the Thurston geometrization application, resulting in the answer of many tough difficulties. a powerful attempt has been made to indicate new connections and views within the box and to demonstrate quite a few points of the idea. An intuitive method which emphasizes the information in the back of the buildings is complemented by means of a number of examples and figures which either use and help the reader's geometric mind's eye.

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Additional info for Conformal Geometry of Discrete Groups and Manifolds (Degruyter Expositions in Mathematics)

Sample text

So we will complete the proof by showing that (1) = (3). To prove that (1) = (3), we first note that the universal covering space X of the manifold X satisfies (2), (3), (4) and (5). 14 because the pair (X, G) satisfies corresponding conditions on transitivity action and compactness of isotropy subgroups in G. Here G is a group of all (X, G)-diffeomorphisms of the universal covering space X which are lifts of elements of G. Therefore, there exists a G-invariant complete Riemannian metric on X. Then we take e > 0 such that the closed ball B(y, e) c X centered at a point y E X is compact, and the transitiveness of the G-action gives us that this is so at any point x E X.

As we saw above, this development d is a local (X, G)-diffeomorphism defined modulo composition with an element of the group G. 38) induces a (X, G)-structure in the universal covering manifold M. Due to uniqueness of the development d (up to composition with g G), we see that each deck transformation Ta E G (M, M) of the universal covering M corresponds to a unique (up to conjugation by g E G) element ga E G such that gad = dTa. 39) which is well defined up to conjugation by an element of the group G.

Each of them is finitely covered by the 3-torus T3 = S1 x S' x St and admits a Seifert fibered space structure. 40), there exist a direction on 1R3 left invariant by r (it is not true for groups with torsion). 17. Let F C Isom R3 be a non-cyclic discrete torsion free group. Then F leaves invariant some family of parallel lines in R3 and the flat manifold II83/ F is Seifert fibered by circles which are the images of these lines. 40)) by conjugation, which induces an action of the finite group H on T.

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Conformal Geometry of Discrete Groups and Manifolds (Degruyter Expositions in Mathematics) by Boris N. Apanasov

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