A note on words having the same image on finite groups

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.