scholarly journals Finitely $\mathcal{F}$-amenable actions and decomposition complexity of groups

2020 ◽  
Vol 14 (3) ◽  
pp. 765-790
Author(s):  
Andrew Nicas ◽  
David Rosenthal
Keyword(s):  
Decomposition Complexity

scholarly journals Coarse property C and decomposition complexity

2017 ◽  
Vol 227 ◽  
pp. 30-50 ◽  
Author(s):  
G. Bell ◽  
D. Moran ◽  
A. Nagórko
Keyword(s):  
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 homology theories and finite decomposition complexity

2019 ◽  
Vol 19 (6) ◽  
pp. 3033-3074 ◽  
Author(s):  
Ulrich Bunke ◽  
Alexander Engel ◽  
Daniel Kasprowski ◽  
Christoph Winges
Keyword(s):  
Decomposition Complexity

Addendum to Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory (J. reine angew. Math. 694 (2014), 129–178)

2019 ◽  
Vol 2019 (746) ◽  
pp. 305-310
Author(s):  
Daniel A. Ramras ◽  
Romain Tessera ◽  
Guoliang Yu
Keyword(s):  
Novikov Conjecture ◽  
K Theory ◽  
Decomposition Complexity

Abstract We supply an argument that was missing from the proof of the main result of the article “Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory” [7]. The argument is essentially formal, and does not affect the strategy of the proof.

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.

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