# Short Laws for Finite Groups and Residual Finiteness Growth

## TL;DR

This paper constructs short laws for finite groups of bounded order and uses these to establish new lower bounds on the residual finiteness growth of non-abelian free groups, advancing understanding of finite group laws.

## Contribution

It introduces shorter laws for finite groups of bounded size and applies these to improve residual finiteness growth bounds for free groups.

## Key findings

- Existence of laws of length n^{2/3} log(n)^{3+δ} for all finite groups of order at most n
- Improved lower bounds on residual finiteness growth of non-abelian free groups
- Enhanced understanding of the relationship between finite group laws and residual finiteness

## Abstract

We prove that for every $n \in \mathbb{N}$ and $\delta>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+\delta}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom, About the length of laws for finite groups, Isr. J. Math.]. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups.