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Groups which are metrically weakly sofic with respect to word norms

Aleksander Ivanov

TL;DR

This work develops a unified metric framework for soficity, LEF, and residual finiteness in groups equipped with word norms, extending the Glebsky–Rivera characterization to normally finitely generated groups and linking these properties to profinite topology. The authors provide a metric analogue of weak soficity for word-normed quotients $\mathsf{F}/N$, culminating in a precise Theorem 4.1 that describes when such normed groups are metrically weakly sofic and applying it to show $ (\mathsf{F}_2, \|\cdot\|)$ is not metrically weakly sofic. They also give corresponding metric LEF characterizations (Theorem 5.1) and explore the relationships between metric RF, LEF, and weak soficity, including conditions under which LEF implies weak soficity and stability results for free groups. The results illuminate how the profinite topology governs metric approximation properties and identify key open questions, such as how these properties extend to higher-rank groups and specific arithmetic groups like $\mathrm{SL}(n,\mathbb{Z})$.

Abstract

We consider metric versions of weak soficity, LEF and residual finiteness. The main results of the paper extend Glebsky and Rivera's characterization of weak soficity to the case of normally finitely generated groups with word metrics. Metric LEF and residual finiteness are also characterized in this class. We deduce that the free group $\mathsf{F}_2$ is not metrically weakly sofic with respect to its standard invariant word norm.

Groups which are metrically weakly sofic with respect to word norms

TL;DR

This work develops a unified metric framework for soficity, LEF, and residual finiteness in groups equipped with word norms, extending the Glebsky–Rivera characterization to normally finitely generated groups and linking these properties to profinite topology. The authors provide a metric analogue of weak soficity for word-normed quotients , culminating in a precise Theorem 4.1 that describes when such normed groups are metrically weakly sofic and applying it to show is not metrically weakly sofic. They also give corresponding metric LEF characterizations (Theorem 5.1) and explore the relationships between metric RF, LEF, and weak soficity, including conditions under which LEF implies weak soficity and stability results for free groups. The results illuminate how the profinite topology governs metric approximation properties and identify key open questions, such as how these properties extend to higher-rank groups and specific arithmetic groups like .

Abstract

We consider metric versions of weak soficity, LEF and residual finiteness. The main results of the paper extend Glebsky and Rivera's characterization of weak soficity to the case of normally finitely generated groups with word metrics. Metric LEF and residual finiteness are also characterized in this class. We deduce that the free group is not metrically weakly sofic with respect to its standard invariant word norm.

Paper Structure

This paper contains 16 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Word norms
  3. Length functions on a group
  4. Standard properties of norms
  5. Metric homomorphisms and almost homomorphisms
  6. Metric RF and metric LEF
  7. Norms for RF and metric LEF
  8. Approximations of finitely generated normed groups with respect to word norms
  9. Plan for Sections 3,4,5
  10. Metric RF
  11. When metric RF does not depend on metric
  12. Metric weak soficity
  13. Metric LEF
  14. Characterization of metric LEF
  15. Connections with metric weak soficity
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $H<G$. For any (invariant) norm $\ell$ (metric $d$) on $G$ the restriction of $\ell$ (resp. $d$) to $H$ is an (invariant) norm (resp. metric).

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 23 more