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

By Gregory C. Bell, Alexander Dranishnikov. 2004

90 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd444de6f7f7afc1aad05ae79c53f0e726c163d3

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Citations in the corpus

Asymptotic Dimension (2007, background): “The proof of the inequality asdimΓ = h(Γ) is given in [8].”
A Hurewicz theorem for the Assouad–Nagata dimension (2006, methodology): “e [18], Theorem 1.12.4 on p.109) says dim(X) ≤ dim(f) + dim(Y) if f: X → Y is a closed map of separable metric spaces and dim(f) is deﬁned as the supremum of dim(f−1(y)), y ∈ Y. Bell and Dranishnikov [3] proved a variant of Hurewicz Theorem for asymptotic dimension without deﬁning the asymptotic dim”
The asymptotic dimension of a curve graph is finite (2005, methodology): “When asdimK and asdim H are both finite, Bell and Dranishnikov proved that asdim G asdim H + asdim K in [3].”
A notion of geometric complexity and its application to topological rigidity (2010, methodology): “l with Z coecients in the variable X2. On the other hand Z=pZoZ, which can be similarly realized as a subgroup of SL(2;Z=pZ[X;X 1]), has nite asymptotic dimension by results of Bell and Dranishnikov [BD3] and Dranishnikov and Smith [DS]. Indeed, it is an extension with both quotient Z and kernel Z=p”
Asymptotic dimension of discrete groups (2006, methodology): “ all R>0. Then asdimΓ ≤ n+ asdimX. Proof. Let F⊂ Γ be a ﬁnitely generated subgroup. Then the R-stabilizer with respect to F is contained in the R-stabilizer with respect to Γ. Then by Theorem 2 of [B-D2], asdimF≤ n+ asdimX. By the deﬁnition, asdimΓ ≤ n+ asdimX. Theorem 2.3 (Hurewicz Type Formula)”
An alternative definition of coarse structures (2006, methodology): “Our final task is to generalize the Hurewicz Theorem for asymptotic dimension of [1] and [3].”
Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, methodology): “Using Wright’s theorem on asymptotic dimension of cube complexes and the Hurewicz-type theorem of Bell and Dranishnikov [3], we obtain the following improved statement.”
Constructing group actions on quasi-trees and applications to mapping class groups (2010, background): “he following theorems. A general reference for asymptotic dimension is [BD08]; for the original deﬁnition and interesting discussion see Gromov’s article [Gro93]. Bell-Dranishnikov’s Hurewicz Theorem [BD06]. Let f : X → Y be a Lipschitz map with X a geodesic space. Suppose that for every R the famil”