Bounded rigidity of manifolds and asymptotic dimension growth

Let X be a metric space and define the category B(X) of manifolds bounded over X in the usual way [FW]. The objects of B(X) are manifolds M equipped with proper2 maps p : M → X. These maps p need not be continuous. A morphism (M1, p1) → (M2, p2) between objects of B(X) is a continuous map f : M1 →M2 which is bounded over X in the sense that there exists k > 0 for which d(p1(m), p2(f(m))) < k for all m ∈ M1. We can additionally form the bounded homotopy category over X by insisting that all relevant maps and homotopies be bounded over the metric space X. Of course, the bounded category is interesting only when X is a space of infinite diameter. IfM is an n-dimensional noncompact manifold, one can construct a structure set Sbdd(M) whose elements are homotopy equivalences M ′ → M bounded over M . As in the compact case, there are bounded L-groups Lbdd ∗ (M) fitting into the exact sequence Hn+1(M ;L(e))→ L n+1(M)→ S(M)→ Hn(M ;L(e))→ L n (M). If Sbdd(M) is trivial, we say that M is boundedly rigid; i.e. if f : M ′ → M is a bounded homotopy equivalence, then f is boundedly homotopic to a homeomorphism. Results of the so-called bounded Borel type can be found in [C], [PW], [FP]. See also [S], [PR], [Ra], [W] and [WW]. One recent theorem of Chang and Weinberger [CW] proves that arithmetic manifolds (those of the form Γ\G/K where Γ is an arithmetic lattice in a real connected linear Lie group G) are boundedly rigid, even though they are in general not properly rigid if the rational rank of Γ exceeds 2. One can view bounded rigidity as topological rigidity in the category of continuous coarsely Lipschitz maps. Let M be a noncompact manifold. We say that M is uniformly contractible if there is a function f : (0,∞) → (0,∞) such that, for each x ∈ X and t > 0, the ball B(x, t) is contractible in the ball B(x, f(t)). This definition arises from the natural notion of the bounded fundamental group in this bounded context. See [W] or [CW] for details. For such uniformly contractible manifolds, Ferry and Pedersen [FP] have shown that two basic principles of surgery theory, the product theorem and the π-π theorem, still hold for bounded surgery problems. It is natural then to consider the rigidity properties of uniformly contractible spaces. In this paper we prove a rigidity theorem for such spaces that satisfy a particular asymptotic dimension condition. The asymptotic dimension of a metric space X is the smallest integer n such that, for any r > 0, there exists a uniformly bounded cover C = {Ci}i∈I for which no ball of radius r in X intersects more than n + 1 members of C. This notion was introduced by Gromov in [G]. As mentioned in [Y1], the notion of asymptotic dimension is a coarse geometric analogue of the covering dimension in topology, invariant under quasi-isometries. It is well known that any finitely generated subgroup of Gromov’s hyperbolic groups has finite asymptotic dimension as metric spaces with word-length metrics. For spaces whose asymptotic dimension is not necessarily finite, we can formulate a more precise notion.