# Download Conformal Geometry of Discrete Groups and Manifolds by Boris N. Apanasov PDF

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 topic is their discrete holonomy teams. particularly, hyperbolic manifolds, in measurement three and better, are addressed. The remedy covers additionally proper topology, algebra (including combinatorial crew idea and sorts of workforce representations), mathematics matters, and dynamics. development in those components has been very speedy sicne the Eighties, particularly as a result of Thurston geometrization application, 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 elements of the idea. An intuitive strategy which emphasizes the information at the back of the structures is complemented via a good number of examples and figures which either use and help the reader's geometric mind's eye.

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Otherwise we may assume that gi (x2) = Y1 for all i . In the latter case, we have an alternative: either yj # xl or yj = xl. 2 where gi (yl) # yi because gi 1(Yi) = x2 ; Yt . If yi = xl, then there exists a third limit point x3 E A(G) \{x2, xt = yj } with images gi (x3) converging to yl. Again, gi (x3) # yj because gi 1(yi) = x2 X3. Thus, in any case, arbitrary limit point yj E A(G) is a limit point of an infinite subset of A(G), and so A(G) is perfect. 44 2. Discontinuous Groups of Homeomorphisms 2.

G is a maximal group of homeomorphisms of the space X with compact stabilizers. In the opposite case, if G were contained in a larger group G', then any (X, G)-manifold would be a (X, G')-manifold, so our (X, G) -geometry would be redundant. This approach has been used by W. 16. Any maximal, simply connected, 3-dimensional geometry which admits a compact quotient is equivalent to one of the eight geometries (X, Isom X) where X is one of R3, H3, S3, If][2 X R, S2 X R, SL2 IR, Nil or Sol. Before we describe these eight geometries, we note that the most interesting, most complicated, and most frequently encountered among them is the hyperbolic one.

We shall describe all elementary groups by the following statement which becomes a criterion in the case of Mobius groups. 4. If a discrete convergence group G C Homeo(S") is virtually Abelian, then G is elementary. , G has an Abelian subgroup of finite index. For the proof we need to classify all cyclic discrete convergence groups. Definition. Let g be a self-homeomorphism of Sn generating a discrete convergence group (g). We say that g is: O elliptic if it is of finite order, ord(g) = inf{m > 0 : gm = id) < oo; (ii) parabolic if ord(g) = oo and g has a unique fixed point; (iii) loxodromic if ord(g) = oo and g has two distinct fixed points.