[Paper Review] Properties of Linearly Sofic Groups

Abel Stolz|arXiv (Cornell University)|Sep 30, 2013

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper introduces and studies linearly sofic groups—groups approximated by matrices over arbitrary fields using rank-based metrics—generalizing sofic and hyperlinear groups. It proves that free products of linearly sofic groups over a fixed field remain linearly sofic, extending a key permanence property from classical sofic group theory to this broader matrix approximation framework.

ABSTRACT

We consider (projectively) linearly sofic groups, i.e. groups which can be approximated using (projective) matrices over arbitrary fields, as a generalization of sofic groups. We generalize known results for sofic groups and groups which can be approximated with complex matrices, including the fact that free products of linearly sofic groups (using a fixed field) are linearly sofic.

Motivation & Objective

• To generalize the theory of sofic groups by introducing linearly sofic groups approximated using matrices over arbitrary fields.
• To investigate whether fundamental permanence properties of sofic groups, such as closure under free products, extend to this broader class.
• To clarify the role of the underlying field's characteristic in group approximation via matrix rank metrics.
• To establish the equivalence between linear and projective matrix approximations in the context of linear soficity.

Proposed method

• Uses pseudo length functions and ultraproduct constructions to define group approximation via matrix sequences over arbitrary fields.
• Employs the rank length ℓr(g) = rk(1−g)/n and Jordan length ℓJ(g) = infα∈K× rk(α−g)/n as approximation metrics.
• Applies amplification techniques to embed group elements into larger matrix representations preserving approximation properties.
• Constructs explicit almost homomorphisms φ: Γ → GL(V) using tensor products and permutation actions on group bases to simulate free product structures.
• Uses ultralimits and metric ultraproducts to analyze asymptotic behavior of matrix sequences and verify approximation conditions.
• Proves equivalence between linear and projective linear approximations by showing that ℓJ captures the same asymptotic behavior as ℓr in the projective setting.

Experimental results

Research questions

• RQ1 Can the class of sofic groups be extended to groups approximated by matrices over arbitrary fields, and what properties do such groups satisfy?
• RQ2 Is the free product of two K-sofic groups itself K-sofic, for a fixed field K?
• RQ3 How does the characteristic of the underlying field affect the approximation of groups using matrix rank metrics?
• RQ4 Are linear and projective matrix approximations equivalent for the class of linearly sofic groups?

Key findings

• The class of K-sofic groups is closed under free products for any field K, generalizing a known result for sofic groups.
• The free product of two K-sofic groups is K-sofic, as shown by constructing explicit almost homomorphisms via tensor product and permutation representations.
• The rank-based metric ℓr and the Jordan length ℓJ yield equivalent notions of approximation, establishing equivalence between linear and projective linear soficity.
• The construction of the almost homomorphism ζ: Γ → GL(V) ensures that ℓJ(ζg) ≥ 1/2 for nontrivial reduced words of length ≤ 2r, guaranteeing separation of identity.
• The method ensures that ℓJ(ζgζh − ζgh) < ε for all g,h in finite subsets, satisfying the almost homomorphism condition with arbitrary precision.
• The proof relies on the fact that cancellations in free products correspond to matching actions in the constructed representation, preserving matrix rank control.