自由熵的秩定理和$L^2$ -不变量：整体上界
The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper Bounds
摘要：
利用微分几何中秩定理的类比，证明了对于迹von Neumann代数中的有限$n$元组$X$和$n$中的$*$ -多项式的任何有限$m$元组$F$，都有\begin{eqnarray*} δ_0(X) & \leq & \text{Nullity}(D^sF(X)) + δ_0(F(X):X) \end{eqnarray*}，其中$δ_0$是（修正的）微观状态自由熵维数，PDMATHAMT是$F$在$X$上的一种导数.当$F(X) =0$和$|D^sF(X)|$具有非零Fuglede-Kadison-Lück行列式时，则$X$在\cite{j3}意义下是$α$有界的，其中$α= \text{Nullity}(D^sF(X))$ .利用Linnell的$L^2$整环在\cite{l}中的结果，以及Elek和Szabó关于Lück关于\cite{es}中sofic群的行列式猜想的工作，证明了以下结果。设$Γ$是一个有2个生成元和$Γ\neq \{0\}$的sofic左可序离散群。以下条件是等效的：（1）$Γ\not\simeq \mathbb F_2$。(2)$L(Γ) \not\simeq L(\mathbb F_2)$。(3)$L(Γ)$是强$1$ -有界的。(4)$δ_0(X) = 1$对于任何生成器的有限集合$X$对于$L(Γ)$。由Brodskiy和Howie关于局部可指示性的结果（\cite{b}，\cite{h}），得出了两个生成元上的无挠单关系子群von Neumann代数是强$1$ -有界的.从正一关系子群（\cite{baum}）的剩余可解性出发，证明了两个生成元上的单关系子群vonNeumann代数如果其关系子是生成元中的非平凡、正、非真字，则它是强$1$ -有界的.
英文摘要：
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*} δ_0(X) & \leq & \text{Nullity}(D^sF(X)) + δ_0(F(X):X) \end{eqnarray*} where $δ_0$ 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-Lück determinant, then $X$ is $α$-bounded in the sense of \cite{j3} where $α= \text{Nullity}(D^sF(X))$. 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 $Γ$ is a sofic, left-orderable, discrete group with 2 generators and $Γ\neq \{0\}$. The following conditions are equivalent: (1) $Γ\not\simeq \mathbb F_2$. (2) $L(Γ) \not\simeq L(\mathbb F_2)$. (3) $L(Γ)$ is strongly $1$-bounded. (4) $δ_0(X) = 1$ for any finite set of generators $X$ for $L(Γ)$. 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.
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