A note on words having the same image on finite groups

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.

💡 Research Summary

The paper investigates the relationship between words in a finitely generated free group when their associated word maps have identical images on every finite group. The central question is whether two such words must be endomorphically equivalent—that is, whether there exist endomorphisms of the free group sending one word to the other and vice‑versa. To address this, the authors introduce the notion of weak profinite rigidity (WPR), a natural analogue of the previously studied profinite rigidity (PR).

A word (w) in a free group (F_n) defines a word map (w\colon G^n\to G) for any group (G); the image of this map is denoted (G_w). Two elements of a finitely generated group are defined to be endomorphically equivalent if each can be obtained from the other by some (not necessarily distinct) endomorphism of the group. Lemma 1.4 (proved via Lemma 2.1) shows that two words have the same image on every group if and only if they are endomorphically equivalent. This result is then lifted to the profinite setting: Theorem 2.4 proves the equivalence among (i) endomorphic equivalence of the images of the elements in the profinite completion (\widehat P), (ii) coincidence of the sets (\operatorname{Hom}_{\gamma,g}(P,G)) for all finite groups (G), (iii) endomorphic equivalence in a suitable finite‑index quotient of (P), and (iv) a uniform condition over all finite‑index normal subgroups.

With this machinery, weak profinite rigidity for a word (w) means that any word (u) sharing the same image on every finite group must be endomorphically equivalent to (w). The authors first observe that any surjective word (i.e., one whose image equals the whole group for every finite group) is automatically weakly profinitely rigid, because surjective words are endomorphically equivalent to each other (Lemma 1.4).

The main contributions are:

1. Powers of primitive words: For any primitive word (p\in F_n) and any integer (d), the word (p^d) is weakly profinitely rigid (Theorem 3.x). This extends the known result that primitive words themselves are profinitely rigid (Puder–Parzanchevski, 2015) to all their powers.
2. Test words: A test word is one that, if fixed by any continuous endomorphism of the profinite completion, must be fixed by an automorphism. The paper shows that test words remain test words in the profinite completion and that for test words, weak profinite rigidity coincides with ordinary profinite rigidity (Theorem 3.y). Consequently, any word sharing the image of a test word on all finite groups also induces the same probability distribution on each finite group.
3. Powers of surface words: Using the above results, the authors prove that the words ((x_1^2\cdots x_n^2)^d) in (F_n) and ((