Free Orbit Dimension of Finite von Neumann Algebras

In this paper we introduce the concept of the upper free orbit-dimension of a finite von Neumann algebra, and we derive some of its basic properties. Using this concept, we are able to improve most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras or simple masas, nonprime factors, those with property $T,$ those with property $\Gamma$, and thin factors. We provide more examples of II$_{1}$ von Neumann algebras whose Voiculesu's free entropy dimension is at most $1$. Our approach gives a strikingly easy way to get to the main results on free entropy dimension, and the paper is nearly self-contained.