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

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 $\ast$-polynomials in $n$ noncommuting indeterminates, \begin{eqnarray*} δ_0(X) & \leq & \text{Nullity}(D^sF(X)) + δ_0(F(X):X) \end{eqnarray*} where ${\delta }_0$ is the (modified) microstates free entropy dimension and $D^{s}F(X)$ is a kind of derivative of $F$ evaluated at $X$. When $F(X)=0$ and $|D^{s}F(X)|$ has nonzero Fuglede-Kadison-Lück determinant, then $X$ is $\alpha$-bounded in the sense of \cite{j3} where $\alpha =\text{Nullity}(D^{s}F(X))$. Using Linnell's $L^2$ integral domain results in \cite{l} as well as Elek and Szabó's work on Lück'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 $\Gamma \ne \{0\}$. 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 Brodskiĭ 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.

67 pages

References in corpus (1)

• Traces on group C*-algebras, sofic groups and Lück's conjecture

Cited by in corpus (2)

• Orthogonal free quantum group factors are strongly 1-bounded
• Strong 1-Boundedness of Unimodular Free Orthogonal Quantum Groups