We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More precisely: The word $w$ is an element of $F_k$, the free group on $k$ letters. The $k$-tuple $\vec{g}$ specifies some homomorphism $\varphi_{ \vec{g} } : F_k \longrightarrow G$ by the universal property. In this notation, $\vec{g}$ satisfies $w$ if $\varphi_{ \vec{g} } (w) = e$.
For a finite group $G$, we are interested in the probability that a random (uniformly chosen) $k$-tuple of group elements satisfies the word $w$. Call this probability $p_w(G)$.
Now consider some family of finite groups, each injecting into the next:
$$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_n \hookrightarrow \ldots $$
Is it true that $p_w(G_n)$ converges to a limit?
For instance, if the word $w$ is $x_1 x_2 x_1^{-1} x_2^{-1}$, then an easy group-theoretic argument shows that $p_w(G_n)$ decreases monotonically.
For the word $x_1^2$, the sequence need not be monotonic, but seems to converge anyway.
Does anyone know a proof that the limit exists? Or have a counterexample?
