The Nikolov-Segal Theorem
In this post I record some notes on the Nikolov-Segal theorem that the topology of a finitely generated profinite group is determined by the algebraic structure, following freely available online resources. Surprisingly, it relies on the classification of finite simple groups. We begin with some definitions.
Definitions: A word is an expression of the form $w(x_1, \dots x_k)= \prod_{j=1}^s x_{i_j}^{\epsilon_j}$ for any choice of $i_1, \dots i_s \in [1,k]$ and $\epsilon_i \in \{ \pm 1\}$.
The length of $w$ is $s$.
For any group $G$ this gives rise to the verbal mapping $w: G^{\times k} \to G$ which sends $(g_1, \dots, g_k)$ (which we will in future abbreviate as $\boldsymbol{g}$) to $\prod_{j=1}^s g_{i_j}^{\epsilon_j}$
Let $w$ be a word and $G$ be a group. The set of $w-$values $G_w$ in $G$ is the set of all possible values of $w(\boldsymbol{g})$ and their inverses. The verbal subgroup of $G$ corresponding to $w$ is group generated by the set of $w-$values.
For example, the verbal subgroup associated to the word $x_1x_2x_1^{-1}x_2^{-1}$ is the commutator subgroup.
Say that a word has width $m$ if $w(G)=G_w^{*m}$, i.e. every element can be written as a product of $m$ elements in $G_w$. The width $m_w(G)$ is the smallest such $m$ for which this holds, or $\infty$ if no such $m$ exists. A group $G$ is said to be verbally elliptic if every word has finite width in $G$. Note that for finite groups $G$ every word has width bounded by $|G|$.
For various reasons, not least intrinsic interest, we would like to find bounds independent of the group order which are true for certain infinite classes of groups.
For a profinite group $G$ denote by $\mathfrak{F}(G)$ the set of finite quotients of $G$ and by $d(G)$ the minimal cardinality of a topological generating set for $G$. It is a fact that $d(G)= \sup \{d(Q) | Q \in \mathfrak{F}(G)\}$, which is one of many instances where the properties of a profinite group and its finite quotients are intimately related.
A profinite group has a topology coming from being a subspace of the direct product of its finite quotients with the discrete topology. With the topology added, finite width gains a topological meaning.
Lemma 1: Let $G$ be a profinite group. Then for each $n \in \Z_{>0}$ the set $G_w^{*n}$ is closed in $G$.
Proof: For each \( n \)-tuple \( (\varepsilon_1, \dots, \varepsilon_n) \in \{ \pm 1 \}^n \), we have a continuous mapping
\[w_{\varepsilon} : G^{\times(nk)} \to G,\]
\[(g_1, \dots, g_n) \mapsto w(g_1)^{\varepsilon_1} \dots w(g_n)^{\varepsilon_n}.\]
Since \( G^{\times(nk)} \) is compact, the image \( X_{\varepsilon} \) of \( w_{\varepsilon} \) is closed in \( G \); hence so is
\[G_w^{*n} = \bigcup_{\varepsilon \in \{ \pm 1 \}^n} X_{\varepsilon}.\]
$\blacksquare$
Proposition 2: For a word $w$ and a profinite group $G$ TFAE:
- there is a finite upper bound for $m_w(Q)$ as $Q$ ranges over $\mathfrak{F}(G)$
- $w$ has finite width in $G$
- $w(G)$ is closed in $G$
Proof: Suppose that
\[m_w(Q) \leq m \quad \text{for all } Q \in \mathfrak{F}(G).\]
Then
\[N G_w^{*(m+1)} = N G_w^{*m} \quad \text{for every } N \triangleleft_o G.\]
Since \( G_w^{*m} \) is closed in \( G \), we obtain
\[G_w^{*m} = \cap N G_w^{*m} = \cap N G_w^{*(m+1)} \supseteq G_w^{*(m+1)}\]
(taking intersections over all \( N \triangleleft_o G \)). It follows that
\[w(G) = G_w^m.\]
Thus, for a profinite group \( G \),
\[m_w(G) = \sup \{ m_w(Q) \mid Q \in F(G) \}.\]
What does this mean topologically? If \( w \) has width \( n \), then \( w(G) = G_w^n \) is closed by Lemma 1. Suppose, conversely, that \( W = w(G) \) is closed. Then the compact Hausdorff space
\[W = \bigcup_{n=1}^{\infty} G_w^{*n}\]
is the ascending union of its closed subspaces \( G_w^{*n} \); it follows by a version of the Baire Category Theorem that for some \( n \), the set \( G_w^n \) contains a non-empty open subset \( U \) of \( W \).
Then \( U \) contains \( hV \) for some \( h \in W \) and some \( V \triangleleft_o W \). In particular, \( V \) has finite index in \( W \), so \( W = YV \) for some finite set \( Y \).
Now \( Y \subseteq G_w^s \) and \( h \in G_w^t \) for some finite \( s \) and \( t \), and we conclude that
\[W \subseteq Y h^{-1} U \subseteq G_w^{(s+t+n)}.\]
Thus, \( w \) has width \( s+t+n \) in \( G \). $\blacksquare$
Let \( C \) be a family of finite groups, closed under forming quotients and finite subdirect products (a ‘formation’). We say that \( w \) is uniformly elliptic in \( C \) if
for each natural number \( d \) there is a natural number \( f_w(d) \) such that
\[H \in C, \quad d(H) \leq d \implies m_w(H) \leq f_w(d).\]
A \pro-\( C \) group is a projective limit of groups in \( C \) (the inverse limit of an inverse system with all maps surjective), or equivalently, it is a profinite group \( G \) such that \( F(G) \subseteq C \).
(When \( C \) is the class of finite \( p \)-groups for some prime \( p \), or the class of finite nilpotent groups, or the class of finite soluble groups, a pro-\( C \) group is said to be pro-\( p \), pronilpotent, or prosoluble, respectively.)
The free pro-\( C \) group on \( d \) generators is the pro-\( C \) completion of \( F_d \), namely the inverse limit
\[(F_d)_C = \lim_{\longleftarrow} (F_d/N),\]
where \( F_d/N \) ranges over all \( C \)-quotients of \( F_d \). Every \( d \)-generator \( C \)-group arises as one of these quotients, so Proposition 2 yields:
Proposition 3: Let \( C \) be a formation of finite groups and let \( w \) be a word. Then \( w \) is uniformly elliptic in \( C \) if and only if the verbal subgroup \( w(G) \) is closed in \( G \) for every finitely generated pro-\( C \) group \( G \).
Definition: Let $d \in \Z_{>0}$. A word $w$ is $d-$ locally finite if $F_d/w(F_d)$ is finite. In other words, $w$ is $d-$locally finite if every $d-$generator group in the variety defined by $w$ is finite.
Proposition 4: Let $G$ be a profinite group. TFAE:
- every subgroup of finite index in $G$
- every normal subgroup of finite index in $G$ is open
- every group homomorphism from $G$ to any profinite group is continuous.
Profinite groups satisfying any of these conditions are said to be strongly complete, since they are their own profinite completion. For such groups the group structure completely determines the topology, which is an extremely strong rigidity phenomenon. The third condition implies that the identity map $G \to G$ is continuous (relative to the original topology on the domain and an arbitrary topology on the range), and every bijective continuous homomorphism of profinite groups is a homeomorphism.
Not all groups are strongly complete, as the following example shows.
Example: Let \( C = <c> \) be a cyclic group of prime order \( p \), and let \( X \) be a countably infinite set. The direct product
\[G = C^X\]
is a profinite group, equipped with the product topology (where \( C \) has the discrete topology). As an abstract group, \( G \) is elementary abelian of uncountable rank \( \mathfrak{c} \), hence there are \( 2^{\mathfrak{c}} \) homomorphisms \( G \to C \). Indeed, given any subset \( U \) of a basis \( T \), we can define a homomorphism by mapping each element of \( U \) to \( c \) and each element of \( T \setminus U \) to \( 1 \).
However, \( G \) has only countably many open subgroups, since each open subgroup contains a basic open subgroup of the form
\[C^{X \setminus Y} \times \prod_Y1\]
for some finite subset \( Y \) of \( X \), and each such basic open subgroup has finite index. It follows that nearly all homomorphisms \( G \to C \) are not continuous.
However, $G$ isn't topologically finitely generated, and as any geometric group theorist will tell you, the world of infinitely generated things is scary. The landmark result that Nikolov-Segal proved is that any finitely generated profinite group is strongly complete, which we now discuss.
Definition: Let $G$ be a finite group and $H$ be a normal subgroup. Then $H$ is acceptable if $H=[H,G]$ and whenever $Z<N\leq H$ are normal subgroups of $G$, the factor $N/Z$ is not of the form $S$ or $S \times S$ for a non-abelian simple group $S$.
Let $\beta(q)(x)=x^q$ be the Burnside word of exponent $q$. The following technical theorem is the main result that they prove:
Theorem 5: Let $G$ be a finite group $H$ an acceptable normal subgroup of $G$.
- [(A)] Suppose that \( G = <g_1, \ldots, g_d> \) and let \( q \in \mathbb{N} \). Then \[ H = (\prod_{i=1}^d [H, g_i])^{\ast f_1(d, q)} \cdot H^{\ast f_2(q)}_{\beta(q)} \]
- [(B)] Suppose that \( G = <g_1, \ldots, g_d> \). Then \[ H = (\prod_{i=1}^d [H, g_i])^{\ast f_3(d)}\cdot H^{\ast D}_{\gamma_2} \]
- [(C)] Suppose that \( d(G) = d \) and that the alternating group \( \text{Alt}(s) \) is not involved in \( G \). If \( G = H <g_1, \ldots, g_r> \), then\[ H = (\prod_{i=1}^r [H, g_i])^{\ast f_4(d, s)}.\]
Here, the functions \( f_1, \ldots, f_4 \) depend only on the displayed arguments, and \( D \) is an absolute constant.
The strategy of proof is as follows: In each case, the theorem asserts the solvability of an equation of the form:
\[w(y_1,...,y_n,g_1,...,g_r)=h\]
where $h \in H$ is the ‘constant’, the ‘unknowns’ $y_1,...,y_n$ are to be found in $H$ and $g_1,...,g_r$ are fixed parameters coming from $G$. The solvability is proved by induction on $|H|$.
Choose $N \trianglelefteq G$ minimal subject to \[1<N=[N,G] \leq H\]. One observes that then $N$ contains a unique normal subgroup $Z$ of $G$ maximal subject to $Z<N$, which implies $N/Z$ is a minimal normal subgroup of $G/Z$. Since $H$ was acceptable, $N/Z$ is either elementary abelian or a direct product of at least 3 isomorphic simple groups.
Now put
\[K = \begin{cases} [Z, G] & \text{if } [Z, G] > 1, \\ N & \text{if } [Z, G] = 1 = N', \\ N & \text{if } [Z, G] = 1 < N'. \end{cases}\]
Applying the inductive hypothesis to \( G/K \), we find \( c \in K \) and \( v_1, \ldots, v_n \in H \) such that
\[c \cdot w(v_1, \ldots, v_n, g_1, \ldots, g_r) = h.\]
To make the induction step, we seek to eliminate the ‘error term’ \( c \) by finding \( a_1, \ldots, a_n \in N \) such that
\[w(a_1v_1, \ldots, a_nv_n, g_1, \ldots, g_r) = h.\]
(This strategy is why the authors say that their proof strategy mirrors that of the proof of Hensel's lemma.)
Writing \( a = (a_1, \ldots, a_n) \) and \( v = (v_1, \ldots, v_n) \), this is equivalent to
\[w'_{v, g}(a, 1) = c. \]
Thus, the problem is to show that \( K \) is contained in the image of the generalized word mapping
\[N^{(n)} \to N \quad \text{defined by} \quad (a_1, \ldots, a_n) \mapsto w'_{v, g}(a, 1).\]
Showing this can actually be solved is the part where the classification of finite simple groups and other results based on the classification come into play.
Corollary 6: Let $d $ and let $w$ be a $d$-locally finite word. Then there exists $f = f(w,d) \in \Z_{>0}$ such that $w$ has width $f$ in every $d$-generator finite group.
Proof: For any $d-$generated group $G$ $w(G)$ is generated by a bounded number of $w-$values $g_1, \cdots ,g_{d'}$. We take $q=|\Z/w(\Z)|$ and choose a suitable characteristic subgroup $H$ of $w(G)$. This automatically satisfies the hypotheses of Theorem 5 while the corollary is already known for the quotient group $G/H$. The result then follows from (A) upon noting that each element $[h,g_i]$ is a product of two $w-$values and each element $h^q$ is a $w$-value. $\blacksquare$
Lemma 7: Let $w$ be a $d-$locally finite word and $G$ a $d-$generator profinite group. Then the closure of $w(G)$ in $G$ is open.
Remark 8: Let $Q$ be a finite group and $d \in \Z_{>0}$. Let $K$ be the intersection of the kernels of all homomorphisms from $F_d$ into $Q$. Let $v_1, \cdots, v_n$ be generators for $K$ and let $w= v_1 \circ \dots \circ v_n$. Then $w$ is a $d-$locally finite word such that $w(Q)=1$.
Theorem 9: Every finitely generated profinite group is strongly complete.
Proof: Let $G$ be a $d-$generator profinite group and $N$ a normal subgroup of finite index. Applying the construction in Remark 8 to $G/N$ gives a word $w$ such that $w(G) \leq N$. The corollary and Proposition 2 show that $w(G)$ is closed. By Lemma 7, it is open. Hence $N$ is open in $G$. $\blacksquare$ Recall that two abstract discrete groups are said to be elementarily equivalent if they satisfy the same sentences in the first order language of group theory. $\blacksquare$
Some applications to words
Lemma 10: Let \( w \) be a \( d \)-locally finite word.
- \( |Z : w(Z)| \) is finite.
- There exists \( \delta \in \mathbb{N} \) such that if \( G \) is any finite \( d \)-generator group, then there exists \( Y \subseteq G_w \) with \( |Y| \leq \delta \) such that \( w(G) = Y \).
Proof: Write \( F = F_d \). Then
\[|Z : w(Z)| \leq |F / w(F)|\]
is finite. Since \( w(F) \) has finite index in \( F \), it is finitely generated, and each generator is a product of finitely many \( w \)-values. Thus, \( w(F) = Z \) for a certain finite subset \( Z \) of \( F_w \). Set \( \delta = |Z| \). $\blacksquare$
Obviously, (ii) holds even if the word ‘finite’ is removed; but here we want to concentrate on the finite case.
Definition: The word \( w \) is \( d \)-restricted if the conclusions (i) and (ii) of lemma above hold.
Theorem 11: Let \( d \in \mathbb{N} \) and let \( w \) be a \( d \)-restricted word. Then there exists \( f = f(w,d) \in \mathbb{N} \) such that \( w \) has width \( f \) in every \( d \)-generator finite group.
A profinite group \( G \) is said to be universal if every finite group occurs as an open section of \( G \); that is, if for every finite group \( H \), there exist open subgroups \( B \triangleleft A \) of \( G \) such that
\[A/B \cong H.\]
We say in this case that \( H \) is involved in \( G \). It is clear that \( G \) is non-universal if and only if the invariant
\[\alpha(G) := \sup \{ n \in \mathbb{N} \mid \text{Alt}(n) \text{ is involved in } G \}= \sup_{Q \in F(G)} \alpha(Q)\]
is finite.
Theorem 12: Let \( q,s \in \mathbb{N} \). The Burnside word \( \beta(q) \) is uniformly elliptic in the class of all finite groups \( G \) with \( \alpha(G) \leq s \).
Both this and Theorem 11 are proved using Theorem 5. The surprising thing is that
Theorem 13: The Burnside word $\beta(q)=x^q$ is $d-$ restricted for every $q$ and every $d$.
This implies that the $d-$restricted words are precisely those that are not commutator words. Together with Theorem 11 this yields
Theorem 14: Every non-commutator word is uniformly elliptic in finite groups.
A model theoretic consequence
Such theorems about words are intimately tied to various rigidity theorems about finitely generated groups, both profinite and discrete. Here we see a lovely consequence due to Lubotzky:
Theorem 15: Let $G$ and $H$ be elementarily equivalent profinite groups. If one of them is finitely generated then they are isomorphic as profinite groups.
This is known as first order rigidity.
Proof: Let $d(G)=d < \infty$ and let $w$ be a $d-$locally finite word. As in the proof of \ref{stronglycomplete} we see that $w(G)$ is open in $G$. Taking $f=f(w,d)$ as in \ref{key} and using the earlier proved fact that $m_w(G)$ is the supremum of $m_w(Q)$ over finite quotients we have that $w(G)=G_w^{*f}$.
Now it is easy to see (but annoying to write out) that the predicate
\[x \in G^{\ast f}_w\]
can be expressed by a formula \( L_{w,f}(x) \) in the language \( L \), and hence so can the statement \( w(G) = G^{\ast f}_w \), which is equivalent to
\[(\forall x \in G) (L_{w,f+1}(x) \Rightarrow L_{w,f}(x)).\]
It follows that \( w(H) = H^{\ast f}_w \).
Next, to each finite group \( F \), we can associate a sentence \( I(F) \) of \( L \) which says that there exist exactly \( |F| \) distinct elements and multiply according to the multiplication table of \( F \).) As a consequence, if \( Q \) is any group, then \( I(F) \) is true in \( Q \) if and only if \( Q \cong F \).
To each sentence \( S \), we associate a sentence \( S_{f,w} \) obtained from \( S \) by replacing each term of the form \( x = y \) by one of the form
\[\exists z (L_{w,f}(z) \wedge x = yz).\]
Thus, if \( K \) is a group such that \( w(K) = K^{\ast f}_w \), then \( S_{f,w}' \) holds in \( K \) if and only if \( S \) holds in \( K/w(K) \). In particular, the sentence
\[I(G/w(G))_{f,w}'\]
is true in \( G \). By assumption it is also true in \( H \), from which it follows that \( H/w(H) \cong G/w(G) \).
Now let \( N \) be any open normal subgroup of \( H \). By Remark 8 there exists a \( d \)-locally finite word \( w \) with \( w(H) \leq N \), and we obtain an epimorphism
\[G \to G/w(G) \cong H/w(H) \to H/N;\]
it is continuous because its kernel contains the open subgroup \( w(G) \). Since the profinite group \( H \) is the inverse limit of all such quotients \( H/N \), it follows that \( H \) is an epimorphic image of \( G \).
Let \( \theta: G \to H \) be an epimorphism, and suppose that \( 1 \neq g \in G \). Then \( g \notin M \) for some open normal subgroup \( M \) of \( G \), and as before we can find a \( d \)-locally finite word \( w \) such that \( w(G) \leq M \).
Now \( \theta \) induces an epimorphism \( \theta^*: G/w(G) \to H/w(H) \); as these groups are isomorphic and finite, \( \theta^* \) is an isomorphism. It follows that
\[\theta (g) \cdot w(H) = \theta^*(g \cdot w(G)) \neq 1\]
since \( g \notin w(G) \). Thus \( \theta(g) \neq 1 \), so \( \theta \) is injective and hence an isomorphism \( G \to H \). $\blacksquare$
One should compare the situation with finitely generated discrete groups, where all finitely generated non-abelian free and surface groups have the same first order theory by Sela's colossal effort, but the situation in the profinite world is completely different. An explanation for this is that profinite groups really should be thought of as built out of finite stuff, and in the proof we showed that finite stuff is first order rigid.
Such theorems about words are intimately tied to various rigidity theorems about finitely generated groups, both profinite and discrete. Here we see a lovely consequence due to Lubotzky:
Theorem 15: Let $G$ and $H$ be elementarily equivalent profinite groups. If one of them is finitely generated then they are isomorphic as profinite groups.
This is known as first order rigidity.
Proof: Let $d(G)=d < \infty$ and let $w$ be a $d-$locally finite word. As in the proof of \ref{stronglycomplete} we see that $w(G)$ is open in $G$. Taking $f=f(w,d)$ as in \ref{key} and using the earlier proved fact that $m_w(G)$ is the supremum of $m_w(Q)$ over finite quotients we have that $w(G)=G_w^{*f}$.
Now it is easy to see (but annoying to write out) that the predicate
\[x \in G^{\ast f}_w\]
can be expressed by a formula \( L_{w,f}(x) \) in the language \( L \), and hence so can the statement \( w(G) = G^{\ast f}_w \), which is equivalent to
\[(\forall x \in G) (L_{w,f+1}(x) \Rightarrow L_{w,f}(x)).\]
It follows that \( w(H) = H^{\ast f}_w \).
Next, to each finite group \( F \), we can associate a sentence \( I(F) \) of \( L \) which says that there exist exactly \( |F| \) distinct elements and multiply according to the multiplication table of \( F \).) As a consequence, if \( Q \) is any group, then \( I(F) \) is true in \( Q \) if and only if \( Q \cong F \).
To each sentence \( S \), we associate a sentence \( S_{f,w} \) obtained from \( S \) by replacing each term of the form \( x = y \) by one of the form
\[\exists z (L_{w,f}(z) \wedge x = yz).\]
Thus, if \( K \) is a group such that \( w(K) = K^{\ast f}_w \), then \( S_{f,w}' \) holds in \( K \) if and only if \( S \) holds in \( K/w(K) \). In particular, the sentence
\[I(G/w(G))_{f,w}'\]
is true in \( G \). By assumption it is also true in \( H \), from which it follows that \( H/w(H) \cong G/w(G) \).
Now let \( N \) be any open normal subgroup of \( H \). By Remark 8 there exists a \( d \)-locally finite word \( w \) with \( w(H) \leq N \), and we obtain an epimorphism
\[G \to G/w(G) \cong H/w(H) \to H/N;\]
it is continuous because its kernel contains the open subgroup \( w(G) \). Since the profinite group \( H \) is the inverse limit of all such quotients \( H/N \), it follows that \( H \) is an epimorphic image of \( G \).
Let \( \theta: G \to H \) be an epimorphism, and suppose that \( 1 \neq g \in G \). Then \( g \notin M \) for some open normal subgroup \( M \) of \( G \), and as before we can find a \( d \)-locally finite word \( w \) such that \( w(G) \leq M \).
Now \( \theta \) induces an epimorphism \( \theta^*: G/w(G) \to H/w(H) \); as these groups are isomorphic and finite, \( \theta^* \) is an isomorphism. It follows that
\[\theta (g) \cdot w(H) = \theta^*(g \cdot w(G)) \neq 1\]
since \( g \notin w(G) \). Thus \( \theta(g) \neq 1 \), so \( \theta \) is injective and hence an isomorphism \( G \to H \). $\blacksquare$
One should compare the situation with finitely generated discrete groups, where all finitely generated non-abelian free and surface groups have the same first order theory by Sela's colossal effort, but the situation in the profinite world is completely different. An explanation for this is that profinite groups really should be thought of as built out of finite stuff, and in the proof we showed that finite stuff is first order rigid.
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