** Dynamic asymptotic dimension and applications, II **

For towers of finite index subgroups, Yamauchi has shown that of box space is either equal to or to .

Let act freely on compact space .

**1. Back to tower dimension **

This is the minimal such that can be -covered by castles, for all finite . A castle is a union of towers. A tower is a disjoint union , open, finite. A castle is a finite collection of disjoint towers. -covered means that every point belongs to some in some tower , and .

This is inspired from Rokhlin towers. It measures to what extent open pieces on which the action is trivial can be made disjoint. Topological dimension is obviously an obstacle.

**2. Nuclear dimension **

** 2.1. Cross-product **

The algebraic cross-product of the action is the complex -algebra generated by and operators , subject to relations , , .

Complete it for the (maximal) norm

** 2.2. Nuclear versus tower dimension **

**Key observation**. Let be a tower and a continuous function on supported on , for some . Then up to isomorphism, the sub--algebra generated by elements , , is contained in the easy -algebra of matrices with entries in .

Let us call -block a -algebra which is a direct sum of full matrix algebras over where ‘s are topological spaces of covering dimensions .

Say a -algebra has finite nuclear dimension if such that for any finite dimensional -subspace and , there are -blocks and contractive completely positive operators such that ,

We now show that, if the action on has tower-dimension and has covering dimension , then the cross-product algebra satisfies the above. We use a “flat” partition of unity , i.e. with small commutators for . This shows

**Theorem 1** * If the action on has finite tower-dimension and has finite covering dimension, then the cross-product algebra has finite nuclear dimension. *

**3. K-theory **

If the -blocks above were ideals, we could compute the K-theory of the cross-product from the K-theory of blocks, by Mayer-Vietoris. But they are not.

Oyono-Oyono and Yu developped a version of K-theory which gives one Mayer-Vietoris-like sequences for subalgebras with the following ideal-like property: for every finite subset , is still a -block. This allows to prove Baum-Connes conjecture and Kunneth formula the cross-product (Guentner-Willett-Yu).

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## About metric2011

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