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Hyperbolic 4-manifolds and tesselations. (English) Zbl 0692.57013
The author generalizes the construction of M. Gromov, B. Lawson and W. Thurston [reviewed above (see Zbl 0692.57012)] to non-trivial plane bundles over a compact surface by putting emphasis on tesselations and discrete actions of groups \(\Gamma_{\nu,n}\subset SO(4,1)\) generated by involutions (see the similar approach in Ch. 9, § 5 of the reviewer’s book “The geometry of discrete groups in space and uniformization problems”, (Kluwer Academic Publ. (Math. Appl. 40), Dordrecht, 1990)). This makes the construction more transparent and yields moduli, in particular rigidity (iff \(\gcd (\nu,n)=1)\) for certain tesselation hyperbolic 4-manifolds \(M^ 4\) and conformal 3-manifolds (at infinity of \(M^ 4)\) which are Seifert fibered manifolds. For a detailed description of the Teichmüller space for such 3-manifolds see K. Ohshika’s paper in Topology Appl. 27, No.1, 75-93 (1987; Zbl 0637.57010).
Reviewer: B.N.Apanasov

MSC:
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
22E40 Discrete subgroups of Lie groups
57S30 Discontinuous groups of transformations
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References:
[1] T. Banchoff, Self-linking numbers of space polygons,Indiana Univ. Math. J.,25 (1976), 1171–1188. · Zbl 0363.53002 · doi:10.1512/iumj.1976.25.25093
[2] T. Banchoff, Normal curvatures and Euler classes for polyhedral surfaces in 4-space,Proc. AMS,92 (1984), 593–596. · Zbl 0558.57010
[3] G. Càlgàreanu, L’intégrale de Gauss et l’analyse des noeuds tridimensionnels,Revue de Math. Pure et Appl.,4 (1959), 5–20.
[4] H. S. M. Coxeter,Introduction to Geometry, New York, Wiley, 1960. · Zbl 0095.34502
[5] Ch.Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable,Colloque de Topologie,cbrm, Bruxelles, 1950, 29–55. Also inOEuvres complètes.
[6] M. Gromov, M. B. Lawson andW. Thurston, Hyperbolic 4-manifolds and conformally flat 3-manifolds. See references there,Publ. Math. I.H.E.S.,68 (1988), 27–45. · Zbl 0692.57012
[7] W. M. Goldman, Projective structures with Fuchsian holonomy,J. Diff. Geom.,25 (3) (1987), 297–326. · Zbl 0595.57012
[8] J. Milnor, On the existence of a connection with zero curvature,Comment. Math. Helv.,32 (1958), 215–233. · Zbl 0196.25101 · doi:10.1007/BF02564579
[9] B. Maskit,Kleinian groups, Grundlehren der Math. Wiss. 287, Springer Verlag (1987). · Zbl 0627.30039
[10] W. Pohl, The self-linking of closed space curves,J. Math. Mech.,17 (1968), 975–985. · Zbl 0164.54005
[11] L. Siebenmann, Regular open neighborhoods, inGeneral topology and its applications,6 (1973), 51–61. See alsoL. Siebenmann, L. Guillou etJ. Hähl, Les voisinages ouverts réguliers,Ann. Sc. École Norm. Sup., 4e sér.,6 (1973), 253–293. · Zbl 0276.57003 · doi:10.1016/0016-660X(73)90030-5
[12] R. Schoen andS. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature,Inv. Math.,92 (1988), 47–72. · Zbl 0658.53038 · doi:10.1007/BF01393992
[13] J. Wood, Bundles with totally disconnected structure group,Comment. Math. Helv.,46 (1971), 257–273. · Zbl 0217.49202 · doi:10.1007/BF02566843
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