Mostow, G.D.: Strong rigidity of locally symmetric spaces. Ann. of Math. Studies 78 Princeton, 1973  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A. The orbits S a are isomorphic to the Furstenberg boundary B = G=P (here P is a minimal parabolic subgroup) for vectors a inside the Weyl chamber, and to quotients of B if a is degenerate [Ka89] The Furstenberg boundary can be also defined as the space of asymptotic classes of Weyl chambers [Mo73] in complete analogy with the definition of the visibility boundary as the space of asymptotic classes of geodesic rays. For the group SL(d; R) the Furstenberg boundary is the space of flags in R (the boundary circle of the hyperbolic plane if d = 2) If a measure on G has a finite first ....

.... non degenerate orbit S a = B; a 2 A 1 , then xn also converges to b 2 B in the Furstenberg compactification [Mo64] Another definition of the Furstenberg boundary B (analogous to that of the visibility boundary S) can be given in terms of maximal totally geodesic flat subspaces of S (flats) [Mo73]. For a given flat f any basepoint x 2 f determines a decomposition of f into Weyl chambers of f based at x. Then B coincides with the space of asymptotic classes of Weyl chambers in S (two chambers are asymptotic if they are within a bounded distance one from the other) 3.4.5. A flat with a ....

G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies, vol. 78, Princeton Univ. Press, Princeton New Jersey, 1973.

....that A = # 0 K xmA . 4.5) Applying condition (C2) we see that x j # 0 for some j, contrary to hypothesis. The result now follows from Lemma 4.1. In the first class of examples, # is the fundamental group of a compact locally symmetric space of nonpositive curvature. The classic book [13] is a convenient reference for the background and necessary results. There is a clear introduction to the theory of symmetric spaces in [2, Chapter II.10] Let X be a symmetric space of noncompact type, by which we mean that X is a quotient G K of a semisimple Lie group by a maximal compact ....

....space W #. It is well known [13, Lemma 4.1] that# may be identified with the topological homogeneous space G P , where P is a Borel subgroup of G. The action of a discrete subgroup # of G on the boundary# will play an important role in our argument, just as it did in Mostow s proof of rigidity [13]. If F is an r flat in X, then the restriction of the equivalence relation to F allows one to define the boundary of F , which is a finite set. There is a natural embedding of the boundary of F into the boundary of X and it is convenient to identify each boundary point of F with the ....

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G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Princeton, New Jersey, 1973.

....polynomial equations. Note that the gluing equations only specify that the angle sum, around each edge or filled cusp, is a multiple of 2. In terms of numerical computation however, it is straightforward to check if a solution actually gives an angle sum of precisely 2. Mostow Prasad rigidity [23] implies that the solution set of the gluing equations is 0 dimensional. It follows that the z i in any solution are algebraic numbers: compare Macbeath s proof of Theorem 4.1 in [17] For example, the complement in S of the figure 8 knot has an ideal triangulation by two tetrahedra with shape ....

G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies 78, Princeton University Press, 1973.

....ffl Complexes with piecewise constant curvature. DJ91, CD95, Ben91, BB94, S93] construct examples with interesting geometric and topological properties. ffl Limits of Hadamard spaces 1 , such as Tits boundaries and asymptotic cones. These have a number of applications, see for example [Mos73, BGS85, KL95, KL97] Supported by a Sloan Foundation Fellowship, and NSF grants DMS 95 05175, DMS 96 26911, DMS 9022140. 1 Following [Bal95] we call CAT (0) spaces (complete simply connected length spaces with nonpositive curvature) Hadamard spaces. 1 In this paper we will study the local ....

G. D. Mostow. Strong rigidity of locally symmetric spaces, volume 78. Ann. of Math. Studies, 1973.

....from [8] 1.1 Introduction The ideal boundary of a locally compact Hadamard space 1 X is a compact metrizable space on which the isometry group of X acts by homeomorphisms. Even though the ideal boundary is a well known construct with many applications in the literature (see for example [10, 4, 2]) the action of the isometry group on the boundary has not been studied closely except in the case of symmetric spaces, Gromov hyperbolic spaces, Euclidean buildings, and a handful of other cases. In the Gromov hyperbolic case 2 the boundary behaves nicely with respect to quasi isometries: any ....

G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, vol. 78.

....munies de m etriques riemanniennes g i localement isom etriques a X i qui varient continument dans la direction transverse. Alors chaque hom eomorphisme Phi : M 1 M 2 qui porte F 1 a F 2 est homotope a un hom eomorphisme Phi 0 : M 1 M 2 affine sur les feuilles. Les th eor emes 1, 2 et [4] impliquent la classification des espaces sym etriques a quasiisometrie pr es: Th eor eme 8 Soient X et X 0 des espaces sym etriques de type non compact. Si X et X 0 sont quasiisom etriques alors ils sont isom etriques apr es renormalisation appropri ee des m etriques sur les facteurs. Les ....

....metrics g i which are locally isometric to X i and which vary continuously in the transverse direction. Then any homeomorphism Phi : M 1 M 2 which carries F 1 to F 2 can be homotoped to a homeomorphism Phi 0 : M 1 M 2 which is affine on leaves. Another consequence of theorems 1 and 2 and [4] is the classification of symmetric spaces up to quasi isometry: Theorem 8 Let X, X 0 be symmetric spaces of noncompact type. If X and X 0 are quasi isometric, then they become isometric after the metrics on their irreducible factors are suitably renormalized. The same theorems provide an ....

G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, vol. 78.

....and (5.8) are also equivalent in the Riemannian setting, provided OE is sense preserving. Recently, the equivalence of (5.7) and (5.8) for homeomorphisms OE : X Y has been established in [44] and [45] for a large class of metric spaces X and Y . For earlier related results, see [78] 60] 43] [72], 66] For historical remarks and further references, see [45] x9. Next we shall apply some of the results obtained in [44] and [45] to our setting. Say that (M; Delta) has locally Q bounded geometry, with Q 1, if there exists a constant C 1 such that each point x 2 M has a neighborhood U so ....

Mostow G. (1974) Strong rigidity of locally symmetric spaces. Annals of Math. Studies, 78, Princeton U.P..

....manifold M to N 1 and N 2 . Thus the map e f 1 # e f 1 2 , where e f 1 2 is any quasi inverse of e f 2 , gives a quasi isometric conjugacy of the two group actions in H 3 . This extends at infinity to a quasi conformal conjugacy of the two group actions on the sphere (see Mostow [73], and in a more general context [34] 27] Furthermore, if the domain of discontinuity is nonempty the equality of ending invariants implies that the map can be made conformal from # 1 to # 2 (on the quotient surfaces there is a bounded homotopy to a conformal map, and this can be lifted to a ....

G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies no. 78, Princeton University Press, Princeton, NJ, 1973.

....started almost twenty years ago, on the applications of harmonic mappings to the study of topology and geometry of Kahler manifolds. The starting point of these developments was the strong rigidity theorem of Siu [1980] which is a generalization of a special case of the strong rigidity theorem of Mostow [1973] for locally symmetric manifolds. Siu s theorem introduced for the first time an e#ective way of using, in a broad way, the theory of harmonic mappings to study mappings between manifolds. Many interesting applications of harmonic mappings to the study of mappings of Kahler manifolds to ....

....attempted to prove Theorem 2.1 by showing that the harmonic map is an isometry. The failure of all these attempts was taken at that time as an indication of the limited applicability of the theory of harmonic maps. In the early 1970 s Mostow proceeded to prove his general rigidity theorem [Mostow 1973], namely the same as Theorem 2.1 with M and N now irreducible compact locally symmetric manifolds, the statement otherwise unchanged. Since what was thought to be the simplest case, namely that of constant curvature manifolds, was not accessible by harmonic maps, no one expected the more general ....

G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton Univ. Press, Princeton, NJ, 1973.

....curvature, normalized so that Vol(M; g 0 ) 1. Then g 0 is the unique entropy minimizing metric up to isometry. Unlike the case of a surface, a locally symmetric metric of negative curvature on a closed orientable n manifold (n 3) is unique up to isometry, by the rigidity theorem of Mostow [17]. A positive solution to the minimal entropy problem appears to single out manifolds that have either a high degree of symmetry or a low topological complexity. What this means in the context of 3 manifolds will become apparent below. A similar phenomena is observed for closed simply connected ....

G. Mostow, `Strong rigidity of locally symmetric spaces', Annals of Mathematics Studies 78, Princeton University Press, 1973.

....that the action is strongly faithful. 12 In case (ii) we use the minimality of the action of Gamma on the sphere at infinity, G=P = G=K) of the symmetric space G=K, where K (respectively P ) denotes a maximal compact (respectively, minimal parabolic) subgroup of G; see Lemma 8. 5 of [Mo73]) The fixed point set in G=P of each element of Gammanf1g is a submanifold of strictly positive codimension, and it thus follows that the action is strongly faithful. Note that in case (ii) if Gamma is cocompact, then it is hyperbolic and G=P is equal to the Gromov boundary, Gamma. Thus if ....

G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Stud., vol. 78, Princeton Univ. Press, 1973.

....(4N Gamma 1) parameter family of solutions to the reduced problem. Our main tool is the study of harmonic maps with prescribed singularities into classical globally symmetric spaces of noncompact and type rank one, see [We4] These are the real , complex , and quaternion hyperbolic spaces, see [Mo]. The results obtained in [We4] however do not apply directly to the problem considered here. In that paper, we restricted our attention to bounded domains Omega ae R n , and to singular sets Sigma compactly contained in Omega Gamma while here Omega = R n and Sigma is unbounded. The ....

G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, 1973.

....proved that a vast collection of 3 manifolds carry metrics of constant negative curvature. These manifolds are thus elements of hyperbolic geometry, as natural as Euclid s regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3 manifolds which can be built up in an inductive way from 3 balls, i.e. Haken manifolds. Thurston s construction of a hyperbolic structure is also inductive. At the inductive step one must find ....

D. Mostow. Strong rigidity of locally symmetric spaces. Annals of Math Studies 78, Princeton University Press, 1972.

....Theorem. The complement of a knot in S 3 admits a hyperbolic structure unless it is a torus or satellite knot. By a hyperbolic structure we mean a complete Riemannian metric of constant sectional curvature 1. Such a metric is uniquely determined by the topology of the knot complement [Mos73, Pra73] and hence is an invariant of the knot itself. In fact, by work of Gromov, Jorgensen, and Thurston, the set of volumes of hyperbolic 3 manifolds is well ordered. Hyperbolic volume is an e ective invariant for distinguishing knots. It distinguishes nearly all hyperbolic knots with up to ....

G.D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, vol. 78, Princeton Univ. Press, 1973.

....classes. Recall that a Kleinian group is a discrete subgroup of the group Isom (H 3 ) of orientation preserving isometries of hyperbolic 3 space. A hyperbolic 3 manifold is the quotient of H 3 by a (torsion free) Kleinian group. It follows from the rigidity theorems of Mostow [9] and Prasad [10] that if N is a 3 manifold that admits a nite volume hyperbolic structure, then the realization of its fundamental group as a Kleinian group is unique up to conjugacy in Isom (H 3 ) Two Kleinian groups 1 and 2 are commensurable if their intersection 1 2 has nite index ....

G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton University Press, Princeton, 1973.

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Mostow, G.D.: Strong rigidity of locally symmetric spaces. Ann. of Math. Studies 78 Princeton, 1973

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G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. Math. Studies, Vol. 78 (1973), Princeton U. Press

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G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. Math. Studies, Vol. 78 (1973), Princeton U. Press

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G.D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, 78, Princeton Univ. Press, 1973.

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Mostow, G. D., Strong Rigidity of Locally Symmetric Spaces (Annals of Math. Studies 78). Princeton Univ. Press, Princeton, 1973.

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G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies, vol. 78, Princeton Univ. Press, Princeton New Jersey, 1973.

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G.D.Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Maths. Studies 78, Princeton U. Press, 1973.

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G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Study, vol. 78, Princeton Univ. Press., 1973.

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G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies 78, Princeton University Press, 1973.

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G. Mostow, Strong rigidity for locally symmetric spaces, Annals of Math., Studies no. 78, Princeton Univ. Press, 1973.

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