Quantifying Residual Finiteness

TL;DR

This paper introduces a method to measure how residually finite certain groups are, providing new insights into their structure and characterizations, especially for nilpotent groups and specific arithmetic groups.

Contribution

It develops a quantitative framework for residual finiteness and offers new characterizations of nilpotent groups within this context.

Findings

Quantification of residual finiteness for various groups

New characterization of nilpotent groups using finite nilpotent quotients

Application to groups like free groups, Grigorchuk group, and $SL_n(\mathbb{Z})$

Abstract

We introduce the concept of quantifying the extent to which a finitely generated group is residually finite. The quantification is carried out for some examples including free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as $SL_n(\mathbb{Z})$. In the context of finite nilpotent quotients, we find a new characterization of nilpotent groups.