Almost Flat Manifolds

1.1. We denote by V a connected ^-dimensional complete Riemannian manifold, by d = d(V) the diameter of V, and by c = c(V) and c~ = c~(V), respectively, the upper and lower bounds of the sectional curvature of V. We set c = c(V) = max (| c1, | c~ |). We say that F i s ε-flat, ε > 0, if cd < ε. 1.2. Examples. a. Every compact flat manifold is ε-flat for any ε > 0. b. Every compact nil-manifold possesses an ε-flat metric for any ε > 0. {A manifold is called a nil-manifold if it admits a transitive action of a nilpotent Lie group; see 4.5.) The second example shows that for n > 3, ε > 0 there are infinitely many ε-flat ^-dimensional manifolds with different fundamental groups. 1.3. Define inductively ext(x) = exp (eXi_λ(x)\ exo(x) — x, and set ε(ή) = exp (—eXj(n)), where j = 200. (We are generous everywhere in this paper because the true value of the constants is unknown.) 1.4. Main Theorem. Let V be a compact έ(n)-flat manifold, and π its fundamental group. Then: (a) There exists a maximal nilpotent normal divisor N C π (b) ord(πlN) 0), then its fundamental group π and every subgroup of π can be generated by 3 elements. (ii) If d(V) < Of, c~(V) > -K,K>0, then π can be generated by N < 3 ex2(nK&) elements; if π is a free group and KQ) 1 < ε(n), then π is generated by one element.