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

TL;DR

This paper establishes new bounds on free entropy dimensions in von Neumann algebras using geometric analogies and $L^2$-invariants, linking algebraic properties of groups to operator algebra invariants.

Contribution

It introduces a rank theorem analogy for free entropy, deriving bounds and equivalences for group von Neumann algebras based on $L^2$-invariants and sofic group properties.

Findings

Equivalence of conditions for sofic, left-orderable groups with 2 generators.

Strong $1$-boundedness for certain group von Neumann algebras.

Relation between free entropy dimension and algebraic group properties.

Abstract

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.