A note on words having the same image on finite groups

TL;DR

This paper investigates when words in free groups are uniquely determined by their images on finite groups, introducing weak profinite rigidity and establishing rigidity results for powers of primitive and surface words.

Contribution

It introduces weak profinite rigidity for words and proves that powers of primitive and surface words are weakly profinitely rigid, linking rigidity to probability measures on finite groups.

Findings

Powers of primitive words are weakly profinitely rigid.

Words with identical images induce the same probability measure on finite groups.

Weak profinite rigidity is equivalent to profinite rigidity for test words.

Abstract

In this work, we explore the following question: If two words in a finitely generated free group have identical images as word maps on every finite group, must they be endomorphic to each other? In this regard, we introduce weak profinite rigidity for words, a parallel to profinite rigidity, as defined in \cite{hanany2020some}. We establish that the powers of primitive words in any finitely generated free group $F_n$ are weakly profinitely rigid. Furthermore, if a word in $F_n$ has the same image on every finite group as a test word in $F_n$, then both words induce the same probability measure on every finite group. We also prove that a test word in $F_n$ is weakly profinitely rigid if and only if it is profinitely rigid. As a consequence, we establish that the powers of surface words, i.e., $(x_1^2\ldots x_n^2)^d$ in $F_n$ and $([x_1,x_2]\ldots [x_{2n-1},x_{2n}])^d$ in $F_{2n}$, for $n \geq 1$ and any integer $d$, are weakly profinitely rigid.