More on words

 In the proof of the Nikolov-Segal theorem many powerful theorems about words were proved. Here we discuss a related theorem of Jaikin-Zapirain, following Segal's book.

Let $H$ denote the free pro$-p$ group on two generators and set $E=H'$.

We quote the following two propositions:

Lemma 1:   Let \( 1 \neq b \in E \) and let \( n \in \mathbb{N} \). Then for some \( d \geq n \), there exists an epimorphism \( \psi: E \to L_d \) with \( \psi(b) \neq 1 \), where \( L_d \) denotes the free pro-\( p \) group on \( d \) generators.

Let \( (L_n)_{n \geq 1} \) denote the dimension subgroup series (‘Zassenhaus filtration’) of the pro-\( p \) group \( L \);

this is the fastest descending chain of open normal subgroups in \( L \) such that

\[[L_n, L] \leq L_{n+1}, \quad L_n^p \leq L^{p_n}\]

for each \( n \). Write \( \pi_n: L \to L/L_n \) for the quotient map. Then the Hausdorff dimension of a subset \( S \) of \( L \) is

\[\mathrm{hdim}(S) = \liminf_{n\to\infty} \mathrm{hdim}_L(\pi_n(S)) = \liminf_{n\to\infty} \frac{\log |S\pi_n|}{\log |L\pi_n|}.\]

Since \( |(S^{\ast t})\pi_n| \leq |S\pi_n|^t \) for each \( n \), we see that

\[\mathrm{hdim}(S^{\ast t}) \leq t \cdot \mathrm{hdim}(S)\]

for each \( t \in \mathbb{N} \).

 Proposition 2: Let \( L \) be the free pro-\( p \) group on \( d \geq 2 \) generators, and let \( N = 1 \) be a closed normal subgroup of \( L \). Say

\[|L : L_{n+1}| = p^{b_n}, \quad |N L_{n+1} : L_{n+1}| = p^{h_n}.\]

Then

\[b_n \sim h_n \sim \frac{d^{n+1}}{(d-1)^n} \quad \text{as } n \to \infty.\]

(Here, \( f(n) \sim g(n) \) means that \( f(n)/g(n) \to 1 \) as \( n \to \infty \).) \label{hdimcor}

 Corollary 3: Let \( L \) be the free pro-\( p \) group on \( d \geq 2 \) generators.

1. If \( N = 1 \) is a closed normal subgroup of \( L \), then \( \mathrm{hdim}(N) = 1 \).
2.  Let \( v \) be the word \( [x,y]^z p \). Then \( \mathrm{hdim}(L^v) \leq \frac{3}{d} \).

Proof: (1): Taking logarithms to base \( p \), we have

\[\frac{\log |N \pi_n|}{\log |L \pi_n|} = \frac{h_{n-1}}{b_{n-1}} \to 1 \quad \text{as } n \to \infty.\]

(2): Since \( L_n \) is central in \( L \) modulo \( L_{n+1} \) and \( L_n^p \leq L^p_n \leq L_{n+1} \), we see that \( L_n \) is marginal for \( v \) modulo \( L_{n+1} \). Therefore, \( v \) takes at most \( |L/L_n|^3 = p^{3b_{n-1}} \) values in \( L/L_{n+1} \). So, putting \( S = L^v \), we have

\[\frac{\log |S \pi_{n+1}|}{\log |L \pi_{n+1}|} \leq \frac{3 b_{n-1}}{b_n} \sim \frac{3n}{d(n-1)} \sim \frac{3}{d} \quad \text{as } n \to \infty.\] $\blacksquare$

\begin{thm} \label{propinfinitewidth}

Theorem 4:  Let $w$ be a word considered as an element of $F_k$. If $1 \neq w \in F''(F')^p$ then $w$ has infinite width in the free pro-$p$ group on two generators.

\end{thm}

Proof: Let \( 1 \neq w \in F/(F^p) \) and suppose that \( w \) has finite width \( m \) in \( H \), the free pro-\( p \) group on two generators. Write \( E = H' \) and for each \( d \), let \( L_d \) denote the free pro-\( p \) group on \( d \) generators.

The hypotheses imply that \( w(H) = W \) is closed and that \( 1 \neq W \leq E' E^p \). In view of Lemma 1, we may choose a large integer \( d \) so that there exists an epimorphism \( \psi: E \to L = L_d \) with \( \psi(W) \neq 1 \).

Since \( w \in F'' (F^p)' = v(F') \), we have

\[w \in (F^v)^{\ast t}\]

for some \( t \). It follows that

\[W = H^{\ast m}_w \subseteq E^{\ast t m}_v,\]

and hence that

\[\psi(W) \subseteq L^{\ast t m}_v.\]

Using corollary 3, we deduce:

\[1=\mathrm{hdim}(\psi(W)) \leq \mathrm{hdim}(L^{\ast t m}_v) \leq t m \cdot \mathrm{hdim}(L_v) \leq \frac{3 t m}{d}.\]

We get a contradiction by choosing \( d > 3 t m \). $\blacksquare$

A word $w \in F_k$ is said to be a $J(p)$ word if $w \notin F''(F')^p$ and for a set $\pi$ of primes it is said to be a $J(\pi)$ word if it is a $J(p)$ word for every $p \in \pi$. 

We now quote a bunch of theorems:

Theorem 5:  Let \( w \) be a word, let \( d \in \mathbb{N} \), and suppose that \( E/w(E) \) is virtually nilpotent where \( E = E_{d+1} \) is the free pro-\( C \) group on \( d+1 \) generators. Then \( w(G) \) is closed in \( G \) for every \( d \)-generator pro-\( C \) group \( G \).

Theorem 6: The following are equivalent:

•  \( w \) is a \( J(p) \) word;
•  \( E/w(E) \) is virtually nilpotent where \( E \) is the free pro-\( p \) group on 2 generators;
• \( G/w(G) \) is virtually nilpotent for every finitely generated pro-\( p \) group \( G \).

Proposition 7: Let \( \Gamma \) be a finitely generated group and \( \pi \) a non-empty set of primes. Suppose that \( \Gamma_p \) is virtually nilpotent for each \( p \in \pi \). Then \( \Gamma_{N(\pi)} \) is virtually nilpotent, and if \( \Gamma \) is residually \( N(\pi) \), then \( \Gamma \) is virtually nilpotent.

Theorem 8: Let \( \pi \) be a non-empty set of primes, and let \( w \) be a \( J(\pi) \) word. Then \( w \) is uniformly elliptic in \( N(\pi) \).

Proof: We have to show that \( w(G) \) is closed in \( G \) for every finitely generated pro-\( C \) group \( G \), where \( C = N(\pi) \). This will follow by Theorem 6 if we prove that \( E/w(E) \) is virtually nilpotent for each finitely generated free pro-\( C \) group \( E \).

Fix \( m \in \mathbb{N} \), set \( F = F_m \), and let \( K/w(F) \) be the \( C \)-residual of \( F/w(F) \), i.e.

\[K = \{N \triangleleft F \mid w(F) \leq N, \, F/N \in N(\pi) \}.\]

Put \( \Gamma = F/K \). For each \( p \in \pi \), the pro-\( p \) completion \( \Gamma_p \) of \( \Gamma \) is an \( m \)-generator pro-\( p \) group, and satisfies \( w(\Gamma_p) = 1 \); it follows by Theorem 5 that \( \Gamma_p \) is virtually nilpotent. Now Proposition 7 shows that \( \Gamma_C \) is virtually nilpotent.

The free pro-\( C \) group \( E = E_m \) on \( m \) generators is the pro-\( C \) completion of \( F \). Therefore,

\[E/w(E) = \lim_{\longleftarrow} \{F/N \mid w(F) \leq N \triangleleft F, \, F/N \in N(\pi) \} = \Gamma_C,\]

and the result follows. $\blacksquare$

Theorem 9:  Let \( \pi \) be a non-empty set of primes and let \( w \) be a non-trivial word. Then the following are equivalent:

•  [(a)] for each \( p \in \pi \), \( w \) has bounded width in 2-generator finite \( p \)-groups;
• [(b)] \( w \) is uniformly elliptic in \( N(\pi) \);
• [(c)] \( w \) is a \( J(\pi) \) word.

Proof:  If (a) holds, then for each \( p \in \pi \), \( w \) has finite width in the 2-generator free pro-\( p \) group, by Proposition 4 of the previous post; this implies (c) in view of Proposition 4. That (c) implies (b) is the content of Theorem 8; and (b) trivially implies (a). $\blacksquare$

The equivalence of (a) and (b) is particularly remarkable: it shows that if the width of \( w \) in two-generator \( p \)-groups is bounded for each \( p \in \pi \), then for each natural number \( m \), it is uniformly bounded in all \( m \)-generator \( p \)-groups, over all \( p \in \pi \).