By P. Hoffman, V. Snaith
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2. A Z * Z covering space. The origins of the line field viewpoint can be found in [Ah5] and [SuI3]. 4 Geometric inflexibility Let M be a hyperbolic 3-manifold whose injectivity radius is bounded above and below in its convex core, but with 8M # 0. Then M need not be rigid; deformations are often possible by changing the conformal structure on 8M. In this section we push the logic of geometric limits further to show a deformation of M decays exponentially fast within its convex core. In other words, the geometry of M deep within the core is inflexible: it changes only a small amount, even under a substantial deformation of 8M.
Suppose the injectivity radius of M in its convex core K ranges in the interval [Ro, Rl], where Ro > 0.. >,p) ~ 1 + Cexp(-ad(p,M - K)). 4. GEOMETRIC INFLEXIBILITY 'fhe constants C and a > 0 depend only on (Ro, Rl' L). Here a bounded homotopy is one which moves points a bounded llyperbolic distance. The maps ~ and \11 are boundedly homotopic if and only if they admit lifts to IHI3 which agree on S~. When K = M, the Theorem says \11 is homotopic to an isometry. ,rhus Mostow rigidity for closed manifolds is a special case, and the ,I'heorem can be thought of as an "effective" version of rigidity for open manifolds.
A shallow set is also small in the sense of dimension. Recall that the box dimension of a compact metric space (X, d) is defined by . (). log N (X, r) box-dIm X = hmsup 1 (1/) , r-+O og r where N (X, r) is the minimum number of r- balls required to cover X. The box dimension is the infimum of those 8 ~ 0 such that N(X,r) = O(r- C). dim(X) is the infimum of those 8 ~ 0 such that X can be covered by balls B(Xi, ri) with E rf arbitrarily small. Clearly H. dim(X) ~ box-dim(X). 21 If A c S~-l is shallow, then the Hausdorff and box dimensions of A are both strictly less than (n - 1).