Alexander Dranishnikov

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

On Asymptotic Dimension of Groups Acting on Trees

We prove the following.THEOREM. Let π be the fundamental group of a finite graph of groups with finitely generated vertex groupsGv having asdim Gv≤nfor all vertices v. Then asdim π≤n+1.This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.

On the Higson corona of uniformly contractible spaces

Abstract Let X be a proper metric space and let νX be its Higson corona. We prove that the covering dimension of νX does not exceed the asymptotic dimension asdimX of X introduced by M. Gromov. In particular, it implies that dim νRn = n for euclidean and hyperbolic metrics on Rn. We prove that for finitely generated groups Γ′ ⊃ Γ with word metrics the inequality dim νΓ′ ⩽ dim νΓ holds. Also we prove that a small action at infinity of a geometrically finite group Γ on some compactification X′ of the universal covering space X = EΓ enables one to map the Higson compactification onto X′. In that case the rational acyclicity of X′ implies the conjecture by S. Weinberger for X which is a form of the Novikov Conjecture for Γ.

On hypersphericity of manifolds with finite asymptotic dimension

We prove the following embedding theorems in the coarse geometry: Theorem A. Every metric space X with bounded geometry whose asymptotic dimension does not exceed n admits a large scale uniform embedding into the product of n + 1 locally finite trees. Corollary. Every metric space X with bounded geometry whose asymptotic dimension does not exceed n admits a large scale uniform embedding into a non-positively curved manifold of dimension 2n + 2. The Corollary is used in the proof of the following. Theorem B. For every uniformly contractible manifold X whose asymptotic dimension is finite, the product X x R is integrally hyperspherical for some n. Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature. We also prove that if a uniformly contractible manifold X of bounded geometry is large scale uniformly embeddable into a Hilbert space, then X is stably integrally hyperspherical.

On intersection of compacta in Euclidean space. II

Suppose that X is a compact subset of n-dimensional Euclidean space Rn. If every map f: y Rnof a compactum Y can be approximated by a map avoiding X then dim X x Y < n .

Extension theory of separable metrizable spaces with applications to dimension theory

The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results: Generalized Eilenberg-Borsuk Theorem. Let L be a countable CW complex. If X is a separable metrizable space and K ∗ L is an absolute extensor of X for some CW complex K, then for any map f : A → K, A closed in X, there is an extension f ′ : U → K of f over an open set U such that L ∈ AE(X − U). Theorem. Let K,L be countable CW complexes. If X is a separable metrizable space and K ∗ L is an absolute extensor of X, then there is a subset Y of X such that K ∈ AE(Y ) and L ∈ AE(X − Y ). Theorem. Suppose Gi, . . . , Gn are countable, non-trivial, abelian groups and k > 0. For any separable metrizable space X of finite dimension dimX > 0, there is a closed subset Y of X with dimGiY = max(dimGiX − k, 1) for i = 1, . . . , n. Theorem. Suppose W is a separable metrizable space of finite dimension and Y is a compactum of finite dimension. Then, for any k, 0 < k < dimW − dimY , there is a closed subset T of W such that dimT = dimW − k and dim(T × Y ) = dim(W × Y )− k. Theorem. Suppose X is a metrizable space of finite dimension and Y is a compactum of finite dimension. If K ∈ AE(X) and L ∈ AE(Y ) are connected CW complexes, then K ∧ L ∈ AE(X × Y ).

ON THE BERSTEIN-SVARC THEOREM IN DIMENSION 2

We prove that for any group π with cohomological dimension at least n the n th power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n : Theorem . For a connected complex X with dim X = cat X = n , we have ≠ 0 where is the Berstein class of X . Previously it was known for n ≥ 3. We also prove that, for every map f : M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.

Transversal intersection formula for compacta

Abstract The main purpose of this paper is to present a unified treatment of the formula for dimension of the transversal intersection of compacta in Euclidean spaces. A new contribution is the proof of inequality dim( X ∩ Y ) ⩾ dim( X × Y ) − n for transversally intersecting compacta X , Y ⊂ R n , based on a correct interpretation of the classical Cogosvili theorem. Also included is a short summary of a new direction of dimension theory, called extension theory, which is needed for the proof.

On macroscopic dimension of rationally essential manifolds

We construct a counterexamples in dimensions

On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

n>3