This is joint work with Ilan Hirshberg.

The two versions of Rokhlin dimension (with and without commuting towers) are numerical invariant for actions of various classes of groups on C*-algebras. Finiteness of these dimensions generalizes the Rokhlin property, which is Rokhlin dimension zero (for both versions). The main use of this concept so far has been that finite Rokhlin dimension is a regularity property: assuming it, one can prove that various properties of C*-algebras are preserved under forming crossed products. However, no attention has been paid to the actual value of the dimension.

In this talk, we consider only the commuting tower version, for finite groups and compact Lie groups. For actions on simple C*-algebras, it was not known whether there are actions of such groups with values other than 0, 1, 2, or infinity. We recently constructed finite group actions on simple AF algebras with arbitrarily large but finite values of Rokhlin dimension, and for every positive integer d, an action of the circle group on a simple AH algebras with no dimension growth which has Rokhlin dimension with commuting towers exactly d. This is thus an interesting invariant, which could be thought of as a measure of the complexity of the action.

In the first part of the talk, I will give the definitions, state some results, and describe some of the constructions; the upper bounds for our examples will follow. In the second part, I will describe equivariant K-theory and how to use it to give lower bounds

on Rokhlin dimension with commuting towers for our examples.