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?



In an inductive family of groups, does the probability that a particular word is satisfied converge?
John, you know this already, and this is far from an answer, but I thought I'd say it here for the benefit of others who may want to think about the problem.


Call a word $w(x\_1,x\_2,\dots,x\_k)$ "groupy" in the variable $x\_i$ if, for fixed values of the other variables, the set of values of $x\_i$ such that $w$ is the identity element is a subgroup of the whole group. Call $w$ "groupy" if it is groupy in all its inputs.


We can show that if $w$ is groupy, and $H \le G$, then $p\_w(H) \ge p\_w(G)$, giving the monotonically decreasing property on $p\_w(G\_n)$.


The word $x$ and the word $e$ are groupy for trivial reasons. Beyond these, the only groupy word I can think of is the commutator word $[x\_1,x\_2]$.


On the other hand, if we restrict ourselves to the variety of abelian groups, all words power words (e.g., $x^2$ or $x^3$) are groupy, hence the monotonically decreasing property holds.


The iterated commutator $[[x\_1,x\_2],x\_3]$ is groupy in $x\_3$ but not (in general) in $x\_1$ or $x\_2$ -- however, it is likely that the groupiness argument can be extended somewhat to cover these kinds of words too.


