Groups which are metrically weakly sofic with respect to word norms

TL;DR

This paper extends the concept of weak soficity to metric groups with word norms, providing characterizations and showing that the free group on two generators is not metrically weakly sofic.

Contribution

It generalizes weak soficity to metric groups with word norms and characterizes metric LEF and residual finiteness within this framework.

Findings

Extends Glebsky and Rivera's characterization to normally finitely generated groups with word metrics

Provides characterizations of metric LEF and residual finiteness

Shows the free group on two generators is not metrically weakly sofic

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.