[Paper Review] Metric Approximations of Wreath Products

Ben Hayes, Andrew W. Sale|arXiv (Cornell University)|Aug 8, 2016

Advanced Topology and Set Theory8 references3 citations

TL;DR

This paper establishes that wreath products preserve soficity, hyperlinearity, linear soficity, and weak soficity under mild conditions: if $H$ is sofic and $G$ is any of these four classes, then $G\wr H$ inherits the same metric approximation property. The authors construct explicit metric approximations using tensor products and cycle decomposition analysis, generalizing prior results on sofic wreath products to broader classes of groups.

ABSTRACT

Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more general metric approximations of groups under wreath products. Our main result is the following. Suppose that $H$ is a sofic group and $G$ is a countable, discrete group. If $G$ is sofic, hyperlinear, weakly sofic, or linear sofic, then $G\wr H$ is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric approximations, extending a general construction of metric approximations for $G\wr H$ that uses soficity of $H$.

Motivation & Objective

• To address the lack of permanence properties for metric approximations of groups, particularly for wreath products.
• To extend known results on sofic wreath products to broader classes: hyperlinear, linear sofic, and weakly sofic groups.
• To construct explicit metric approximations for $G\wr H$ when $H$ is sofic and $G$ belongs to one of the four classes.
• To prove that the metric approximation properties are preserved under wreath products through a unified, quantitative framework.

Proposed method

• A general construction of metric approximations for $G\wr H$ is developed, relying on the soficity of $H$ and the structure of permutation actions on sets.
• For linear soficity, the authors use almost homomorphisms into $\mathrm{GL}_n(\mathbb{F})$ with rank metric control, and analyze the behavior of tensor products of operators.
• The key technical tool is a formula for the rank metric of the image of a wreath product map, derived from cycle decomposition of permutations and kernel dimension analysis.
• The proof uses a decomposition of the kernel of $\Psi(A,\tau) - \mathrm{Id}$, relating it to fixed points and cycles of $\tau$, leading to a lower bound on $\ell_{\mathrm{rk}}(\Psi(A,\tau))$.
• Bi-invariance and metric comparison techniques are used to establish almost multiplicativity and almost injectivity of the approximating maps.
• The construction is applied to $\mathrm{GL}_n(\mathbb{F})\wr_{B}\mathrm{Sym}(B)$ and $\mathrm{PGL}_n(\mathbb{K})\wr_{B}\mathrm{Sym}(B)$, with length functions defined via rank metrics.

Experimental results

Research questions

• RQ1 Does the wreath product $G\wr H$ preserve soficity when $H$ is sofic and $G$ is sofic?
• RQ2 Can the permanence property be extended to hyperlinear, linear sofic, and weakly sofic groups under the same conditions?
• RQ3 How can metric approximations for $G\wr H$ be constructed explicitly when $G$ is not necessarily amenable or residually finite?
• RQ4 What role do cycle decompositions and tensor products play in maintaining metric control in the approximations?
• RQ5 Is the rank metric behavior under tensor products stable enough to preserve linear soficity in wreath products?

Key findings

• If $H$ is sofic and $G$ is sofic, then $G\wr H$ is sofic, generalizing Paunescu's result for abelian $G$.
• If $H$ is sofic and $G$ is hyperlinear, then $G\wr H$ is hyperlinear, extending the class of hyperlinear groups.
• If $H$ is sofic and $G$ is linear sofic, then $G\wr H$ is linear sofic, with the proof relying on rank metric control under tensor products.
• If $H$ is sofic and $G$ is weakly sofic, then $G\wr H$ is weakly sofic, via a general construction of metric approximations.
• The authors derive a precise formula for $\ell_{\mathrm{rk}}(\Psi(A,\tau))$ in terms of the Hamming distance of $\tau$ and the rank metrics of $A_b$, showing that $\ell_{\mathrm{rk}}(\Psi(A,\tau)) \geq \frac{1}{2}\ell_{\mathrm{Hamm}}(\tau) + \frac{1}{|B|}\sum_{\tau(b)=b} \ell_{\mathrm{rk}}(\Phi(A_b))$.
• The construction ensures $(F, \varepsilon, d_{\mathrm{rk}})$-multiplicativity and $(F, c'/2, d_{\mathrm{rk}})$-injectivity of the approximating map $\Psi \circ \Theta$, proving the desired metric approximation property.