# Flexible stability and nonsoficity

## TL;DR

This paper explores the concept of flexible stability in sofic groups, showing that if certain linear groups are flexibly stable, then non-sofic groups must exist, challenging existing assumptions in group theory.

## Contribution

It introduces the notion of flexible stability for sofic groups and links it to the potential existence of non-sofic groups, providing new insights into group approximation properties.

## Key findings

- If PSL_d(Z) is flexibly stable for some d ≥ 5, then non-sofic groups exist.
- Flexible stability relates to the structure of Schreier graphs and group approximations.
- The paper establishes a conditional connection between stability properties and the existence of non-sofic groups.

## Abstract

A sofic group $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if $\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for some $d \geq 5$ then there exists a group which is not sofic.