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.
