How to mention about group words when you haven't said about free groups and show regarding a marginal set computation Let $F$ be a free group on a countably infinite set $\{x_1, x_2, \ldots \}$ and let $W$ be a nonempty subset of $F$. If $w = w(x_1, \ldots , x_n) \in W$ and $g_1, \ldots, g_n$ are elements of a group $G$, we define the value of the word $w$ at $(g_1, \ldots, g_n)$ to be $w(g_1, \ldots, g_n)$.
When speaking about a group $G$, say a semidirect product $G=H \rtimes_{\phi} K$, where $H$ and $K$ are finite subgroups and $H$ is the normal subgroup, or any group, we usually haven't specified about a free group but just talk about the group. There are values that can be written in terms of elements of the group, like,
$w_1 = h_1^2 h_2^3$, where $h_1, h_2$ are generating elements of the group,
$w_2 = g_1^3 g_2 g_3^2$, where $g_1,g_2,g_3 \in G$ etc.
So, $w_1, w_2$ represent values (or elements) of the group $G$. But the expressions $h_1^2 h_2^3, g_1^3 g_2 g_3^2$ are like words. Can they be called group words?
(According to definition in wikipedia, "in group theory, a word is any written product of group elements and their inverses"
https://en.wikipedia.org/wiki/Word_(group_theory)#:~:text=In%20group%20theory%2C%20a%20word,group%20elements%20and%20their%20inverses.&text=Two%20different%20words%20may%20evaluate,study%20in%20combinatorial%20group%20theory.)
Since, we originally didn't specify a free group how can I speak about these expressions.
Even though I haven't mentioned a free group, can I say like $w_1$ is a group word in terms of $x_1, x_2$ and I'm considering the value of $w_1$ at $(h_1,h_2)$? (and then use that value to compute marginal sets as mentioned in this previous post :Meaning of Marginal sets and related descriptions)
I mean usually when talking about a group since we don't start by mentioning about a free group how can I mention about the words and values of the words used for marginal set computation? Just talk about the value instead of words?
Thanks a lot in advance.
 A: The underlying issue here seems to be that there are two definitions of "words", which are related but different. I will call your definition a word map, and the definition from Wikipedia a group word. This phrasing seems to be consistent with the literature on word maps*, although in the broader group theory setting a "group word" is usually just called a "word".
Let $S$ be a set, called an alphabet, and let $S^{-1}$ be the set of formal inverses of elements of $S$. Let $\mathbf{x}\in(S\sqcup S^{-1})^n$ be an $n$-tuple.

*

*If $S$ is additionally a subset of $G$, then a word map $w$ over $S$ is a function from $(S\cup S^{-1})^n$ to $G$, $w: (S\cup S^{-1})^n\rightarrow G$. The value of $w$ at $\mathbf{x}$ is the group element $w(\mathbf{x})$.


*If $S$ is equipped with an embedding $\phi:S\hookrightarrow G$, then a group word $W$ over $S$ is an element of the free monoid $(S\cup S^{-1})^*$. Contrasting with the above, $W(\mathbf{x})$ is group word, and is not a group element but instead represents a group element (namely $W(\phi(\mathbf{x}))$, defined appropriately).
So we can see that, roughly speaking, the difference is about when we place the "word" into the group. For a group word this is done at the last step: first we form $W(\mathbf{x})$, and then we embed it into the group via $\phi$. For a word map, $S$ is placed in the group at the beginning and so everything takes place in the group.
The expressions $h_1^2h_2^3$ and $g_1^3g_2g_3^2$ from your question are group words, as they merely represent group elements. They are not "values", but instead represent values. Another group word is $h_1^2h_2^3h_1h_1^{-1}$, which represents the same group element as $h_1^2h_2^3$.

Lets dig into that phrase "defined appropriately", in the definition of a group word. I'll mention two ways of doing this.
Firstly, we can use word maps. Indeed, word maps and group words are related because a group word $W$ defines a word map $\overline{W}:(\phi(S)\cup \phi(S)^{-1})^n\rightarrow G$, and so the group word $W(\mathbf{x})$ represents the value $\overline{W}(\phi(\mathbf{x}))$.
Secondly, we can use free groups. In the above I never mentioned the word "free group", but the embedding $\phi:S\hookrightarrow G$ extends to a homomorphism $\widehat{\phi}:F(S)\rightarrow\langle \phi(S)\rangle$. This is a second way to "appropriately define" $W(\phi(\mathbf{x}))$.

*See for example the paper Larsen, Michael, and Aner Shalev. "Word maps and Waring type problems." Journal of the American Mathematical Society 22.2 (2009): 437-466. (doi, arXiv).
