** Dynamic asymptotic dimension and applications, I **

With Guentner and Yu.

**1. Motivations **

1. Controlled topology. In 1998, Yu showed that if a group has a finite classifying space and finite asymptotic dimension, then Novikov conjecture holds.

2. Structure of -algebras. In 2010, Winter and Zacharias showed that the nuclear dimension (a dimension theory for -algebras) of the Uniform Roe Algebra of group is less than or equal to its asymptotic dimension.

**Goal**: find a notion of dimension for actions of groups on compact spaces which has similar consequences.

**2. Dynamic asymptotic dimension (DAD) **

This is the smallest such that space has a a -element open cover such that within each open set, the orbit equivalence relation has bounded classes.

**Examples**.

- The action of on its Stone-\v Cech compactification has DAD .
- Finite DAD implies finite stabilizers. Therefore, from now on, we shall assume that actions are free.
- actions have DAD (Putnam).
- actions on finite dimensional spaces have DAD between and (Szabo).
- nilpotent group actions on finite dimensional spaces have finite DAD (Szabo-Zacharias).
- Given a tower of finite index subgroups, the action on the corresponding completion has DAD equal to the asdim of the box space.
- Finite DAD implies action is amenable.

**Question**. Can DAD take other values than and ?

**3. Tower dimension **

This alternate notion, due to David Kerr, is more handy for applications. It is bounded above and below by DAD, hence both are simultaneously finite or infinite. It also satisfies .

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/