Arbitrarily Large Residual Finiteness Growth

TL;DR

This paper constructs groups with arbitrarily large residual finiteness growth and explores its connection to decision problems in group theory, providing new insights into group approximation properties.

Contribution

It introduces a method to create groups with unbounded residual finiteness growth and links this property to decision problems in groups.

Findings

Groups with arbitrarily large residual finiteness growth are constructed.

A new relationship between residual finiteness growth and decision problems is demonstrated.

Applications to understanding group approximation and decision complexity.

Abstract

The residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. In this paper, we construct groups with arbitrarily large residual finiteness growth. We also demonstrate a new relationship between residual finiteness growth and some decision problems in groups, which we apply to our new groups.