Locally symmetric spaces are generalizations of spaces of constant curvature. In this book the author presents the proof of a remarkable phenomenon, which he calls "strong rigidity": this is a stronger form of the deformation rigidity that has been investigated by Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.
The proof combines the theory of semi-simple Lie groups, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian spaces, elements of ergodic theory, and the fundamental theorem of projective geometry as applied to Tit's geometries. In his proof the author introduces two new notions having independent interest: one is "pseudo-isometries"; the other is a notion of a quasi-conformal mapping over the division algebra K (K equals real, complex, quaternion, or Cayley numbers). The author attempts to make the account accessible to readers with diverse backgrounds, and the book contains capsule descriptions of the various theories that enter the proof.
*Frontmatter, pg. i*Contents, pg. v*
1. Introduction, pg. 1*
2.
Algebraic Preliminaries, pg. 10*
3. The Geometry of chi : Preliminaries, pg.
20*
4. A Metric Definition of the Maximal Boundary, pg. 31*
5. Polar Parts,
pg. 35*
6. A Basic Inequality, pg. 44*
7. Geometry of Neighboring Flats, pg.
52*
8. Density Properties of Discrete Subgroups, pg. 62*
8. Density
Properties of Discrete Subgroups, pg. 66*
10. Pseudo Isometries of Simply
Connected Spaces with Negative Curvature, pg. 71*
11. Polar Regular Elements
in Co-Compact GAMMA, pg. 76*
12. Pseudo-Isometric Invariance of Semi-Simple
and Unipotent Elements, pg. 80*
13. The Basic Approximation, pg. 96*
14. The
Map , pg. 103*
15. The Boundary Map 0, pg. 107*
16. Tits Geometries, pg.
120*
17. Rigidity for R-rank > 1, pg. 125*
18. The Restriction to Simple
Groups, pg. 128*
19. Spaces of R-rank 1, pg. 134*
20. The Boundary
Semi-Metric, pg. 142*
21. Quasi-Conformal Mappings Over K and Absolute
Continuity on Almost All R-Circles, pg. 156*
22. The Effect of Ergodicity,
pg. 169*
23. R-Rank 1 Rigidity Proof Concluded, pg. 180*
24. Concluding
Remarks, pg. 187*Bibliography, pg. 193*Backmatter, pg. 197
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