The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper Bounds

Using an analogy with the rank theorem in differential geometry, it is shown that for a finite $n$-tuple $X$ in a tracial von Neumann algebra and any finite $m$-tuple $F$ of $*$-polynomials in $n$ noncommuting indeterminates, \begin{eqnarray*} \delta_0(X) & \leq & \text{Nullity}(D^sF(X)) + \delta_0(F(X):X) \end{eqnarray*} where $\delta_0$ is the (modified) microstates free entropy dimension and $D^sF(X)$ is a kind of derivative of $F$ evaluated at $X$. When $F(X) =0$ and $|D^sF(X)|$ has nonzero Fuglede-Kadison-L\"uck determinant, then $X$ is $\alpha$-bounded in the sense of \cite{j3} where $\alpha = \text{Nullity}(D^sF(X))$. Using Linnell's $L^2$ integral domain results in \cite{l} as well as Elek and Szab\'o's work on L\"uck's determinant conjecture for sofic groups in \cite{es} the following result is proven. Suppose $\Gamma$ is a sofic, left-orderable, discrete group with 2 generators and $\Gamma \neq \{0\}$. The following conditions are equivalent: (1) $\Gamma \not\simeq \mathbb F_2$. (2) $L(\Gamma) \not\simeq L(\mathbb F_2)$. (3) $L(\Gamma)$ is strongly $1$-bounded. (4) $\delta_0(X) = 1$ for any finite set of generators $X$ for $L(\Gamma)$. From Brodski\u{i} and Howie's results on local indicability (\cite{b}, \cite{h}), it follows that a sofic, torsion-free, one-relator group von Neumann algebra on two generators with nontrivial relator is strongly $1$-bounded. It also follows from the residual solvability of the positive one relator groups (\cite{baum}) that a one-relator group von Neumann algebra on two generators whose relator is a nontrivial, positive, non-proper word in the generators is strongly $1$-bounded.