00 the space (V*, g~) converges over each v E V to the corresponding M-bundle over IRk = Tv(V), that is the homogeneous space M+ = Aff(lRk)/ maxcomp = IRk X M where this (natural) splitting is invariant under Aff IRk, the affine automorphism group of IRk. Notice that this convergence may be (for k ;:: 2) non-uniform in m E M albeit M is homogeneous. In fact, if k ;:: 2 the metric g~ for each), may easily have unbounded curvature on a fiber Mv.

Then take the second such hypersurface (in the complement V-H 2 ), say H 2 , and continue by transfinite induction thus arriving at a closed set Ho = C£ U Hi C V with the following properties. iEI 1. If V - Ho for some leaf V has a component with more than one end, then the closure of this component contains some Hi C Ho. Or, equivalently, a slightly moved Hi separates ends in this components. 2. If Hi and H j have mutually E-close points then Hi is Hausdorff 8-close to H j for some 8 = 8(10) -> 0, for 10 -> 0.

Is recdim V :S k here as well? g. universal) coverings compact manifolds V as follows. V of Define dim(V IV) as the minimal number k, such that V can be cov~ed by k + 1 open subsets Ui , i = 0, ... , k, where each conne~ted lift of Ui to V is relatively compact. ) Now one observes that this dim VIV bounds the macroscopic dimension of V (at least for Galois coverings where "finite-to-one" has "finite" :S const) and one asks oneself if the opposite is true. Here one has the famous Stallings' theorem about ends of groups which refines the implications dime; V :S 1 => dim VIV :S 1 as follows.