ar X iv : m at h / 99 07 14 3 v 1 [ m at h . SG ] 2 2 Ju l 1 99 9 THE SYMPLECTIC GEOMETRY OF POLYGONS IN HYPERBOLIC 3 - SPACE

ar X iv : m at h / 04 07 44 1 v 1 [ m at h . SG ] 2 6 Ju l 2 00 4 THE GEOMETRY OF SYMPLECTIC PAIRS

We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.

ar X iv : m at h / 99 07 08 5 v 1 [ m at h . G R ] 1 3 Ju l 1 99 9 LOOPS AND SEMIDIRECT PRODUCTS

ar X iv : m at h / 99 07 18 5 v 1 [ m at h . M G ] 2 9 Ju l 1 99 9 Higher dimensional flexible polyhedra ∗

It is show that sphere homeomorphic flexible polyhedra (with self intersections) do really exist in n-dimensional Euclidean, Lobachevskij and spherical spaces for each n ≥ 3. 1991 Mathematics Subject Classification: 52C25, 52B11.

ar X iv : m at h / 99 07 13 9 v 1 [ m at h . G T ] 2 2 Ju l 1 99 9 CONSTRUCTING HYPERBOLIC MANIFOLDS

The Coxeter simplex with symbol is a compact hyperbolic 4-simplex and the related Coxeter group Γ is a discrete subgroup of Isom(H 4). The Coxeter simplex with symbol is a spherical 3-simplex, and the related Coxeter group G is the group of symmetries of the regular 120-cell. Using the geometry of the regular 120-cell, Davis [3] constructed an epimorphism Γ → G whose kernel K was torsion-free, ...

ar X iv : m at h / 99 07 13 7 v 1 [ m at h . L O ] 2 2 Ju l 1 99 9 STRONGLY MEAGER AND STRONG MEASURE ZERO SETS

In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets.