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 $w(g_1, \ldots, g_n)$.
If $W$ is a non-empty set of words in $x_1, x_2, \ldots$ and $G$ is any group, a normal subgroup $N$ is said to be $W$-marginal in $G$ if, $w(g_1, \ldots, g_n) = w(u_1 g_1, \ldots , u_n g_n)$, for all $w(x_1, \ldots , x_n) \in W$, $g_i \in G$, $u_i \in N$, $1 ≤ i ≤ n$.
This is equivalent to the requirement: $g_i = f_i\pmod N$, $1 ≤ i ≤ n$, always implies that $w(g_1, \ldots , g_n) = w(f_1, \ldots , f_n)$.
The following definition extends the marginality property. (suppose $W$ consists of a single word $w$. it can be extended to any $W$).
Definition 1: For $n \in \mathbb{N}$, let $w = w(x_1, \ldots , x_n)$ be a group word, $G$ be a group and $\bar{g} = (g_1, \ldots, g_n)$ be a tuple of elements of $G$. We say that a tuple $\bar{c} = (c_1,\ldots , c_n) \in G^n$ is a marginal tuple determined by $w$ and $\bar{g}$ if, $w(c_1g_1, \ldots, c_ng_n) = w(g_1, \ldots , g_n)$.
We will write $\bar{c} \perp w(\bar{g})$ in this case ($\bar{c}$ is a marginal tuple with respect to $w$ and $\bar{g}$). A set $\bar{C} \subseteq G^n$ is said to be marginal with respect to $w$ and $\bar{g} (\bar{C} \perp w(\bar{g}))$ if, $\bar{c} \perp w(\bar{g})$ for every tuple $\bar{c} \in \bar{C}$.
According to the above descriptions,
Q1: When we say "This is equivalent to the requirement: $g_i = f_i\pmod N$, $1 ≤ i ≤ n$, always implies that $w(g_1, \ldots , g_n) = w(f_1, \ldots , f_n)$", can someone please help me to understand this more clearly?
$g_i = f_i\pmod N$ means that $g_i f_i^{-1} \in N$, right, when taking mod by a group?
When we have $w(g_1, \ldots, g_n) = w(u_1 g_1, \ldots , u_n g_n)$, then is it like, $g_i = u_i g_i \pmod G$?
Q2: According to the definition, what is meant by "The $i^{th}$ component $c_i$ of any marginal tuple $\bar{c}$ can be any element of $G$ if $w$ does not in fact depend of $x_i$?.
What is the meaning of $w$ does not in fact "depend" of $x_i$?
Thanks a lot in advance.
P.S. the above descriptions and definitions are from https://www.researchgate.net/profile/Vitaly-Romankov/publication/332480171_An_improved_version_of_the_AAG_cryptographic_protocol/links/608646048ea909241e262a48/An-improved-version-of-the-AAG-cryptographic-protocol.pdf
Question 1: $x\equiv y\pmod{N}$ holds if and only if $xN=yN$, if and only if there exist $n,n'\in N$ such that $x=yn$ and $xn'=y$; if and only if $Nx = Ny$; if and only if there exist $n,n'\in N$ such that $x=ny$ and $y=n'x$. In particular, if $g_i\equiv f_i\pmod{N}$, then there exist $u_1,\ldots,u_n\in N$ such that $g_iu_i=f_i$. Thus, $w(g_1,\ldots,g_n)=w(f_1,\ldots,f_n)$ follows from the marginality of $N$. Conversely, if whenever $g_i\equiv f_i\pmod{N}$ we have $w(g_1,\ldots,g_n)=w(f_1,\ldots,f_n)$, then for all $u_1,\ldots,u_n\in N$ we have $u_ig_i\equiv g_i\pmod{N}$, and hence $w(u_1g_1,\ldots,u_ng_n)=w(g_1,\ldots,g_n)$, proving that $N$ is $w$-marginal.
Question 2: The word $w$ does not "in fact depend on $x_i$" if the value of $w$ on a tuple does not depend on the value of $x_i$. For example, consider the word $w(x,y) = xyx^{-1}$. If $G$ is abelian, then the value of $w$ at a pair of elements of $G$ does not in fact depend on $x$, since $w(a,b)=b$ for all $a,b\in G$. So the "value" of $x$ does not affect the value of $w(x,y)$. That is, the value of $w(x,y)$ does not depend on what $x$ is.