By Joachim Cuntz, Georges Skandalis, Boris Tsygan

This quantity includes contributions by means of 3 authors and treats elements of noncommutative geometry which are on the topic of cyclic homology. The authors provide particularly whole bills of cyclic concept from diverse and complementary issues of view. The connections among topological (bivariant) K-theory and cyclic concept through generalized Chern-characters are mentioned intimately. This contains an overview of a framework for bivariant K-theory on a class of in the community convex algebras. however, cyclic conception is the ordinary atmosphere for a number of basic index theorems. A survey of such index theorems (including the summary index theorems of Connes-Moscovici and of Bressler-Nest-Tsygan) is given and the recommendations and concepts thinking about the evidence of those theorems are defined.

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L l2 l3 Fig. 20. P l P Q 1 .. .... .. ... . 1 . . . . . . . . . . . . . . ..... 1 .... .... . .. 2.................................................................. .......... 3 .. .. ... .. .. 8. Isometries as projective transformations Viewed in projective geometry, an isometry φ of H 2 extends naturally to a projective transformation Φ of the projective plane P 2 associated to H 2 , namely, a bijective self-mapping Φ of P 2 that sends any projective line to a projective line.

If the two straight lines are perpendicular, then the conclusion is obviously true. Otherwise, suppose the two straight lines form an acute angle P OQ with P Q ⊥ OQ. 9, as P moves in ray OP away from O, the distance |P Q| increases. Now take a point P1 on ray OP and produce OP1 to a sequence of points P2 , · · · , Pn , · · · on ray OP so that |OP1 | = |P1 P2 | = |P2 P3 | = · · · = |Pn−1 Pn | = · · · . For each n = 1, 2, · · · , draw Pn Qn ⊥ OQ with foot Qn on OQ. 7, we have |P2 Q2 | > 2|P1 Q1 |, and inductively, |P2n Q2n | > 2n |P1 Q1 | for n = 1, 2, · · · .

41. In a hyperbolic plane an equidistant curve is not a straight line. Hint: The proof is similar to that for a horocycle or a circle. 42. A generalized circle in a hyperbolic plane is either a circle, a horocycle, or an equidistant curve. The pencil consisting of all the straight lines perpendicular to a generalized circle is called its radii pencil. 43 (Circumcircle Theorem). Through the three vertices of a triangle in a hyperbolic plane passes a unique generalized circle, called the generalized circumcircle of the triangle.