Approximability of word maps by homomorphisms

TL;DR

This paper proves that approximate word maps in finite groups can be closely approximated by genuine homomorphisms, providing explicit bounds and constructions, with potential applications in group theory.

Contribution

The paper introduces an explicit function and construction showing how approximate word maps can be approximated by homomorphisms, generalizing previous results.

Findings

Existence of an explicit function relating approximation quality to homomorphism proximity

Construction of a reduced word v in at most 3d variables approximating w

Identification of a large fiber in the word map v_G for groups with approximate homomorphism behavior

Abstract

Generalizing a recent result of Mann, we show that there is an explicit function $f:\left(0,1\right]\rightarrow\left(0,1\right]$ such that for every reduced word $w$, say in $d$ variables, there is an explicit reduced word $v$ in at most $3d$ variables (nontrivial if the length of $w$ is at least $2$) such that for all $\rho\in\left(0,1\right]$, the following holds: If $G$ is any finite group for which the word map $w_G:G^d\rightarrow G$ agrees with some fixed homomorphism $G^d\rightarrow G$ on at least $\rho|G|^d$ many arguments, then the word map $v_G:G^{3d}\rightarrow G$ has a fiber of size at least $f(\rho)|G|^{3d}$. We also discuss some applications of this result.