By G. Daniel Mostow

ISBN-10: 0691081360

ISBN-13: 9780691081366

Locally symmetric areas are generalizations of areas of continuous curvature. during this e-book the writer offers the facts of a outstanding phenomenon, which he calls "strong rigidity": this can be a superior type of the deformation tension that has been investigated by means of Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.

The facts combines the speculation of semi-simple Lie teams, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian areas, parts of ergodic conception, and the basic theorem of projective geometry as utilized to Tit's geometries. In his facts the writer introduces new notions having autonomous curiosity: one is "pseudo-isometries"; the opposite is a idea of a quasi-conformal mapping over the department algebra okay (K equals genuine, complicated, quaternion, or Cayley numbers). the writer makes an attempt to make the account obtainable to readers with assorted backgrounds, and the publication includes pill descriptions of a few of the theories that input the proof.

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**Example text**

2 . This formula is familiar in form, if not in meaning. It looks just like the formula expressing the co-associativity of the co-multiplication in a Hopf algebra (or even in a co-algebra). Coincidences like this are not "accidents" in mathematics. What is happening here is that every Hopf algebra is a left co-module over itself with the co-action being the co-multiplication. One more word of caution: Each of the double indices in ! 10/ and ! 3). 2). It is ! D ".! 0/ /! 4) for all ! 2 . Again, this is reminiscent of a formula for Hopf algebras.

We will see later how this concept comes back to the commutative setting of finite groups in Chapter 13 to give us a new way of introducing infinitesimal structures there. 2, we have seen for the first time a q-deformation. These give many, though by no means all, examples of objects in noncommutative geometry. We will comment more on q-deformations in the notes at the end of Chapter 12. 1 Definitions We now assume that A is a Hopf algebra. The co-multiplication in A is denoted by W A ! A ˝ A; and the co-inverse (also known as the antipode) is denoted by Ä W A !

A for which we have the following two commutative diagrams: ˆ ! ? A ? 7) ˆ˝id ! ˝A˝A and ˆ ? A ? 8) Š ! ˝ C: One also says that a vector space together with a given right co-action ˆ is a right A-co-module. 1. 8). 2. As with almost all definitions, there are trivial examples. v/ WD 1 ˝ v for all v 2 . Prove that T is indeed a left co-action. Then define the trivial right co-action of A on . There is some other terminology that is used for left and right co-actions, and this has been known to lead to a lot of avoidable confusion.

### Strong Rigidity of Locally Symmetric Spaces by G. Daniel Mostow

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