scholarly journals Asymptotic dimension, decomposition complexity, and Haver's property C

2014 ◽  
Vol 169 ◽  
pp. 99-107 ◽  
Author(s):  
Alexander Dranishnikov ◽  
Michael Zarichnyi
Keyword(s):  
Asymptotic Dimension ◽  
Dimension Decomposition ◽  
Decomposition Complexity

scholarly journals Burghelea conjecture and asymptotic dimension of groups

2018 ◽  
Vol 12 (02) ◽  
pp. 321-356
Author(s):  
Alexander Engel ◽  
Michał Marcinkowski
Keyword(s):  
Cyclic Homology ◽  
Asymptotic Dimension ◽  
Group Rings ◽  
Finitely Generated ◽  
Type Formula ◽  
Complex Group ◽  
Bass Conjecture ◽  
Thompson’S Group ◽  
Decomposition Complexity

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type [Formula: see text]) counterexample based on Thompson’s group [Formula: see text]. We construct as well a finitely generated counterexample with finite decomposition complexity.

scholarly journals COARSE AMENABILITY AND DISCRETENESS

2015 ◽  
Vol 100 (1) ◽  
pp. 65-77 ◽  
Author(s):  
JERZY DYDAK
Keyword(s):  
The Other ◽  
Asymptotic Dimension ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Property A ◽  
Main Concept ◽  
Other Hand ◽  
Necessary And Sufficient ◽  
Decomposition Complexity

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

scholarly journals Regular finite decomposition complexity

2019 ◽  
Vol 11 (03) ◽  
pp. 691-719 ◽  
Author(s):  
Daniel Kasprowski ◽  
Andrew Nicas ◽  
David Rosenthal
Keyword(s):  
Asymptotic Dimension ◽  
Finite Quotient ◽  
Permanence Properties ◽  
Decomposition Complexity

We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov’s finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one called Finite Quotient Permanence. We show that for a collection containing all metric families with finite asymptotic dimension, all other permanence properties follow from Fibering Permanence.

Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane

2010 ◽  
Vol 53 (4) ◽  
pp. 629-638
Author(s):  
Naotsugu Chinen ◽  
Tetsuya Hosaka
Keyword(s):  
Asymptotic Dimension

AbstractIn this paper, we investigate a proper CAT(0) space (X, d) that is homeomorphic to ℝ2 and we show that the asymptotic dimension asdim(X, d) is equal to 2.

scholarly journals On equivariant asymptotic dimension

2017 ◽  
Vol 11 (3) ◽  
pp. 977-1002 ◽  
Author(s):  
Damian Sawicki
Keyword(s):  
Asymptotic Dimension

scholarly journals A note on spaces of asymptotic dimension one

2007 ◽  
Vol 7 (2) ◽  
pp. 1063-1070 ◽  
Author(s):  
Koji Fujiwara ◽  
Kevin Whyte
Keyword(s):  
Asymptotic Dimension ◽  
Dimension One
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