Read-back: fg_and_polynomial_growth_of_finiteIndex

One declaration is rendered here. It is a single theorem whose conclusion is a three-fold conjunction.

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Preliminary: the notions the statement is built from

These have to be spelled out, because the statement's content is entirely carried by them. Each is quoted in full so that nothing is hidden behind a name.

Word balls

Let $\Gamma$ be a group and let $S \subseteq \Gamma$ be any subset. For a natural number $n$, the word ball of radius $n$ is

$$B_S(n) \;=\; \bigl\{\, g \in \Gamma \;\bigm|\; \text{there is a finite list } (x_1,\dots,x_r) \text{ of elements of } \Gamma \text{ with } r \le n,\; \text{each } x_i \in S \text{ or } x_i^{-1} \in S,\; \text{and } x_1 x_2 \cdots x_r = g \,\bigr\}.$$

Four features of this are worth making explicit.

• The condition on each letter is $x_i \in S$ or $x_i^{-1} \in S$; so the letters are drawn from $S \cup S^{-1}$, and $S$ itself is not required to be symmetric.
• The length condition is $r \le n$, not $r = n$; the ball is closed, containing all shorter words as well.
• The empty list is allowed, with the convention that the empty product is the identity. Consequently $1 \in B_S(n)$ for every $n$, including $n = 0$; and $B_S(0) = \{1\}$.
• The product is formed in the order the letters are listed, folding from the right: $x_1 \cdot (x_2 \cdot (\cdots (x_r \cdot 1)))$. By associativity this is just the ordered product $x_1 x_2 \cdots x_r$.

The growth function

For a finite subset $S$ of $\Gamma$ and a natural number $m$,

$$\beta_S(m) \;=\; \#\, B_S(m),$$

the number of elements of the word ball, as a natural number. The counting function used here returns $0$ for an infinite set (a junk convention); this does not bite, since $S$ is finite and therefore every $B_S(m)$ is finite.

Polynomial growth of degree at most $E$

For a group $\Gamma$ and a natural number $E$, the assertion "$\Gamma$ has polynomial growth of degree $\le E$" means:

there exist a finite subset $S \subseteq \Gamma$ that generates $\Gamma$ (the subgroup generated by $S$ is all of $\Gamma$), and a real number $c$ with $c > 0$, such that for every natural number $m$ with $m \ge 1$,

$$\beta_S(m) \;\le\; c \cdot m^{E},$$

the left side being read as a real number.

Note the shape of the quantifiers: the generating set is existentially quantified, so the assertion is that some finite generating set satisfies the bound, not that every one does. In particular the assertion already entails that $\Gamma$ is finitely generated. Note also that $m = 0$ is excluded from the bound, and that $E$ is a natural number, so that for $E = 0$ the right-hand side is the constant $c$ and the assertion is that the word-ball sizes are bounded.

The lower central series and its ranks

For a group $\Gamma$, write $\gamma_0(\Gamma), \gamma_1(\Gamma), \gamma_2(\Gamma), \dots$ for the lower central series indexed from zero:

$$\gamma_0(\Gamma) = \Gamma, \qquad \gamma_{k+1}(\Gamma) = [\gamma_k(\Gamma), \Gamma],$$

where $[A,B]$ denotes the subgroup generated by the set of commutators $aba^{-1}b^{-1}$ with $a \in A$, $b \in B$. (So the indexing is offset by one from the convention in which the series starts at $\gamma_1 = \Gamma$.)

For each $k$, form the following abelian group. Take $\gamma_k(\Gamma)$ as a group in its own right and abelianize it, i.e. pass to

$$\gamma_k(\Gamma) \big/ [\gamma_k(\Gamma), \gamma_k(\Gamma)].$$

Inside $\gamma_k(\Gamma)$, consider the subgroup consisting of those elements of $\gamma_k(\Gamma)$ that also lie in $\gamma_{k+1}(\Gamma)$, take its image under the abelianization map, and quotient by that image. Call the resulting abelian group

$$Q_k(\Gamma) \;=\; \Bigl( \gamma_k(\Gamma)\big/[\gamma_k(\Gamma),\gamma_k(\Gamma)] \Bigr) \Big/ \operatorname{image}\bigl( \gamma_k(\Gamma) \cap \gamma_{k+1}(\Gamma) \bigr).$$

(The construction is the literal one just described; no identification of $Q_k$ with $\gamma_k/\gamma_{k+1}$ is asserted by the statement, even where such an identification holds.)

Now define

$$r_k(\Gamma) \;=\; \operatorname{rank}_{\mathbb Z} Q_k(\Gamma),$$

meaning: regard $Q_k(\Gamma)$ additively (its addition is the group operation of $Q_k(\Gamma)$, and the integer $n$ acts on an element $x$ by $n \cdot x = x^n$), which makes it a $\mathbb{Z}$-module; take the supremum of the cardinalities of its $\mathbb{Z}$-linearly independent subsets; and convert that cardinal to a natural number, with the convention that an infinite cardinal is sent to $0$. Concretely, $r_k(\Gamma)$ is the largest $r$ for which there are elements $x_1,\dots,x_r$ of $Q_k(\Gamma)$ such that $x_1^{n_1}\cdots x_r^{n_r} = 1$ forces $n_1 = \cdots = n_r = 0$ — the torsion-free rank — and $r_k(\Gamma) = 0$ when no finite such largest $r$ exists. In particular $r_k(\Gamma) = 0$ whenever $Q_k(\Gamma)$ is a torsion group (for instance whenever it is finite), and also $r_k(\Gamma) = 0$ when the torsion-free rank is infinite.

The nilpotency class and the exponent $e_2$

For a group $\Gamma$, the nilpotency class $c(\Gamma)$ is the least natural number $n$ such that the $n$-th term of the upper central series of $\Gamma$ is all of $\Gamma$ — the upper central series being $Z_0 = 1$, $Z_1 =$ the centre, and $Z_{n+1} = \{x : [x,y] \in Z_n \text{ for all } y\}$. If $\Gamma$ is not nilpotent, $c(\Gamma)$ takes the junk value $0$. For the trivial group, $Z_0 = 1 = \Gamma$ already, so $c(\Gamma) = 0$.

Finally, define the natural number

$$e_2(\Gamma) \;=\; \sum_{k=0}^{c(\Gamma)-1} 2^{k} \, r_k(\Gamma),$$

i.e. the sum over $k = 0, 1, \dots, c(\Gamma)-1$ (an empty sum, hence $0$, when $c(\Gamma) = 0$).

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The statement

Fix

• a type $G$ in an arbitrary universe, carrying a group structure;
• the hypothesis that $G$ is finitely generated: some finite subset of $G$ generates $G$;
• a subgroup $H \le G$ (this argument is explicit; $G$ and its group structure are implicit, and the finite-generation hypothesis and the next one are supplied as typeclass assumptions);
• the hypothesis that $H$ has finite index in $G$: the number of left cosets of $H$ in $G$, counted as a natural number with the convention that an infinite index is recorded as $0$, is nonzero. (Equivalently: $G/H$ is a finite set.)

Throughout, $H$ is regarded as a group in its own right — the underlying set is $\{g \in G : g \in H\}$ with the multiplication inherited from $G$ — and every statement about "$H$" below is a statement about that group. In particular the lower central series, the nilpotency class and the generating sets appearing in the assertions about $H$ are all computed inside $H$, not in $G$.

Then all three of the following hold simultaneously (the conclusion is a conjunction of three claims).

(1) $H$ is finitely generated. There is a finite subset of $H$ which generates $H$ as a group, i.e. whose generated subgroup of $H$ is the whole of $H$.

(2) Polynomial growth transfers from $H$ up to $G$, degree for degree. For every natural number $E$: if $H$ has polynomial growth of degree $\le E$, then $G$ has polynomial growth of degree $\le E$. Spelled out: if there are a finite subset $T \subseteq H$ generating $H$ and a real $c > 0$ with $\#B_T(m) \le c\, m^{E}$ for all integers $m \ge 1$ (the balls taken inside $H$), then there are a finite subset $S \subseteq G$ generating $G$ and a real $c' > 0$ with $\#B_S(m) \le c'\, m^{E}$ for all integers $m \ge 1$ (the balls taken inside $G$). This is an implication in one direction only; nothing is asserted about the converse.

(3) If $H$ is nilpotent, the degree can be replaced by the minimum of $E$ and $e_2(H)$. Assume in addition that $H$ is nilpotent (some term of its upper central series is all of $H$). Then for every natural number $E$: if $H$ has polynomial growth of degree $\le E$, then $G$ has polynomial growth of degree $\le \min\bigl(E,\, e_2(H)\bigr)$, the minimum taken among natural numbers, where

$$e_2(H) \;=\; \sum_{k=0}^{c(H)-1} 2^{k}\, r_k(H)$$

is the quantity defined above, computed for the group $H$: $c(H)$ is the nilpotency class of $H$, and $r_k(H)$ is the torsion-free rank (with the $0$-for-infinite convention) of the abelian group

$$\Bigl( \gamma_k(H)\big/[\gamma_k(H),\gamma_k(H)] \Bigr) \Big/ \operatorname{image}\bigl( \gamma_k(H) \cap \gamma_{k+1}(H) \bigr),$$

with $\gamma_0(H) = H$ and $\gamma_{k+1}(H) = [\gamma_k(H), H]$ computed inside $H$.

The three claims are asserted jointly, for the single fixed $H$; claims (2) and (3) each quantify over all natural numbers $E$ separately, inside the conclusion.

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Degenerate and boundary cases the quantifiers include

• $E = 0$. Both (2) and (3) include $E = 0$. There the hypothesis says the word-ball sizes of $H$ are bounded above by a constant, and the conclusion says the same for $G$.

• $\min$ can be $0$. In (3), $\min(E, e_2(H))$ is $0$ whenever $e_2(H) = 0$, which happens in particular when $c(H) = 0$ (the sum is empty — e.g. $H$ trivial) and also when every $r_k(H) = 0$ (e.g. when all the abelian groups $Q_k(H)$ are finite). In those cases (3) asserts that $G$ has polynomial growth of degree $\le 0$, i.e. that the word-ball sizes of $G$ are bounded by a constant.

• $\min$ vs. $E$. Since $\min(E, e_2(H)) \le E$, claim (3) is a statement about an exponent no larger than the one in claim (2). The statement asserts the bound with the exponent $\min(E, e_2(H))$ exactly; it does not assert any equality, nor that $e_2(H)$ is the least possible exponent, nor anything about $e_2(H)$ itself beyond its appearance inside the minimum.

• Junk-value conventions in force. Three total functions with junk values appear: the cardinality of an infinite set is $0$ (harmless here, the word balls being finite); the nilpotency class of a non-nilpotent group is $0$ (in (3) it is guarded by the nilpotency hypothesis, so it is the genuine class there); and a $\mathbb{Z}$-rank that is infinite is recorded as $0$, which is not guarded by anything and is a genuine part of what $r_k(H)$ means.

• Vacuity. None of the hypotheses is impossible to satisfy: the trivial group with its trivial subgroup satisfies all of them, and so does any finite group with any subgroup. The antecedents inside (2) and (3) ("$H$ has polynomial growth of degree $\le E$", "$H$ is nilpotent") are ordinary implications: if they fail for a given $H$ and $E$, the corresponding claim holds without content for that $H$ and $E$, but claim (1) is asserted unconditionally.

• No symmetry or minimality is required of the generating sets. In each occurrence of "polynomial growth of degree $\le E$", the generating set is a fresh existential; the sets witnessing the hypothesis in $H$ and the conclusion in $G$ need not be related, and neither is required to be symmetric — symmetry is built into the ball instead.