Universal spaces for asymptotic dimension

Dranishnikov, Alexander; Zarichnyi, Mykhailo

doi:10.1016/j.topol.2003.07.009

Cited by 59 publications

(88 citation statements)

References 14 publications

(23 reference statements)

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“…In [14] X satisfying this condition are said to have asymptotic dimension ≤ n of linear type. It is shown in [10] that every metric space of bounded geometry with asdim X ≤ n admits a coarsely equivalent metric with the Higson property. Unfortunately the coarse type of H(X) depends on a metric on X, not only on the coarse class of metrics.…”

Section: Proof Take G = O(n 1) + and K = O(n)mentioning

confidence: 99%

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

Bell

Dranishnikov

2006

Trans. Amer. Math. Soc.

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Abstract. 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 finitedimensional 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.

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Section: Proof Take G = O(n 1) + and K = O(n)mentioning

confidence: 99%

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

Bell

Dranishnikov

2006

Trans. Amer. Math. Soc.

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“…The existing proof in Bell, Dranishnikov and Keesling [5] is quite long and it appeals to the asymptotic inductive dimension theory developed by the author and Zarichnyi [17].…”

Section: Introductionmentioning

confidence: 99%

On asymptotic dimension of amalgamated products and right-angled Coxeter groups

Dranishnikov

2008

Algebr. Geom. Topol.

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“…well-known in the Asymptotic Topology as the asymptotic counterpart of the Cantor set, see [4], [7].…”

Section: A Simple Example Of a Dual Pair Of Micro And Macro Fractalsmentioning

confidence: 99%

Micro and Macro Fractals Generated by Multi-Valued Dynamical Systems

Банах

Novosad

2014

Fractals

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Abstract. Given a multi-valued function Φ : X X on a topological space X we study the properties of its fixed fractal Φ , which is defined as the closure of the orbit Φ ω ( * Φ ) = n∈ω Φ n ( * Φ ) of the set * Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals Φ and Φ −1 for a contracting compact-valued function Φ : X X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

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Universal spaces for asymptotic dimension

Abstract: Topology and its Applications 140 (2004) 203-225. doi:10.1016/j.topol.2003.07.0092016-03-04T18:46:49

Cited by 59 publications

References 14 publications

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

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

On asymptotic dimension of amalgamated products and right-angled Coxeter groups

Micro and Macro Fractals Generated by Multi-Valued Dynamical Systems

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