## Notes of Rufus Willett’s first Southampton lecture 28-03-2017

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 ${C^*}$-algebras. In 2010, Winter and Zacharias showed that the nuclear dimension (a dimension theory for ${C^*}$-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.

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

Examples.

• The action of ${G}$ on its Stone-\v Cech compactification ${\beta G}$ has DAD ${=asdim(G)}$.
• Finite DAD implies finite stabilizers. Therefore, from now on, we shall assume that actions are free.
• ${{\mathbb Z}}$ actions have DAD ${=1}$ (Putnam).
• ${{\mathbb Z}^m}$ actions on finite dimensional spaces have DAD between ${m}$ and ${3^m}$ (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 ${asdim}$ and ${+\infty}$?

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 ${tower-dim(\beta G)=asdim(G)}$.