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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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