Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$C∗-algebras

By E. Guentner, Rufus Willett, Guoliang Yu. 2015

99 citations recorded by the source corpus.

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov’s theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for $$C^*$$C∗-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3c4ea62df588a9002428ba37e4d24b187ff67009

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Jianchao Wu
2. David Kerr
3. G'abor Szab'o
4. J. Zacharias
5. Hanfeng Li
6. Xiangdong Ye
7. Eli Glasner
8. Frederic Pierre Jean Latrémolière

In Mathematical Physics

1. G'abor Szab'o
2. David Kerr
3. Jianchao Wu
4. J. Zacharias
5. W. Winter
6. Hanfeng Li
7. Ben Hayes
8. Xiangdong Ye

In Statistical and Nonlinear Physics

1. G'abor Szab'o
2. David Kerr
3. Jianchao Wu
4. J. Zacharias
5. W. Winter

Citations in the corpus

• Almost Finiteness and the Small Boundary Property (2018, background): “have long played an important role in unraveling the structure of C∗-crossed products, notably in the case of Z-actions [35, 26, 25, 39, 13] and in the various theories of dimension for group actions [18, 36, 37, 21], the novelty in the concept of almost 1The first proof of this fact was given by Con”
• Rokhlin dimension for actions of residually finite groups (2014, background): “ \ σ . Then Ug·σ= g·Uσand Uσ∩Uσ′ = ∅ for σ6= σ′ of the same dimension. Now U is defined to be the collection U = {Uσ | σ∈ Z}, which gives us an open cover of |Z| of multiplicity at most d+1. 8In [20], the terminology “(M,ε)-equivariant” is used instead. ROKHLIN DIMENSION FOR ACTIONS OF RESIDUALLY ”