Appendix: Gromov’s polynomial growth theorem

The aim of this homework is to prove the following theorem due to Misha Gromov in 1981.

Theorem 5.3 Let \(G\) be a finitely generated group of polynomial growth then \(G\) has a finite index subgroup nilpotent group.

We say that a group with a finite index nilpotent group is virtually nilpotent, so the theorem says that a group with polynomial growth is virtually nilpotent.

This is a converse to the fact that any finitely generated nilpotent group has polynomial growth (see lecture notes).

This homework is essentially filling the details in the blog post of Terry Tao where Terry Tao explains a more elementary proof of this theorem than the original one. It relies on a deep result of Bruce Kleiner that will be stated below but that we won’t prove. The following sections are quite independent. Only the last one puts everything together.

5.6 About the other direction

Question 5.1 Let \(G\) be group and \(H\) a subgroup of finite index. Prove that \(G\) is finitely generated if and only if \(H\) is finitely generated.

Question 5.2 Prove that a finitely generated group \(G\) and a finite index subgroup \(H\) are quasi-isometric.

Question 5.3 Let \(G\) be a finitely generated group and \(H\) a finite index subgroup. Prove that \(G\) has polynomial growth if and only if \(H\) has polynomial growth.

Question 5.4 Let \(G\) be a finitely generated nilpotent group. The goal of this hypothesis is to prove that its commutator subgroup \(G'=[G,G]\) is finitely generated. Let \(S\) be a finite generating set of \(G\).

Prove that any commutator can be written as a product of iterated commutators \([\dots[[s_1,s_2],s_3],\dots,s_n]\) with \(s_i\in S\). (Hint: use iteratively the relation \(xy=yx[x^{-1},y^{-1}]\)).
Prove that there is a finite quantity of such iterated commutators of elements of \(S\).
Conclude that \(G'\) is finitely generated.

Question 5.5 Let \(G\) be the lamplighter group \[\left(\oplus_{\mathbf{Z}}\mathbf{Z}/2\mathbf{Z}\right)\rtimes\mathbf{Z}\] where \(\oplus_{\mathbf{Z}}\mathbf{Z}/2\mathbf{Z}\) is the set of biinfinite sequences \((a_i)_{i\in\mathbf{Z}}\) where \(a_i\in \mathbf{Z}/2\mathbf{Z}\) and \(a_i=0\) for all but finitely many \(i\)’s. The group \(\mathbf{Z}\) acts on \(\oplus_{\mathbf{Z}}\mathbf{Z}/2\mathbf{Z}\) via \(n\cdot(a_i)=(a_{i-n})\).

Prove that the lamplighter group is a finitely generated solvable group
Prove that its derived subgroup \(G'\) is not finitely generated
Explained why the proof that finitely generated nilpotent group have polynomial growth does not work for solvable groups.

Question 5.6 Give an example of a group of polynomial growth that is not nilpotent.

5.7 The space of Lipschitz harmonic functions

Let \(G\) be a finitely generated group with finite symmetric generating set \(S\). The fact that \(S\) is symmetric means that \(\forall s\in S, s^{-1}\in S\).

Definition 5.4 Let \(f\colon G\to \mathbf{R}\). This function is harmonic if for any \(g\in G\):

\[f(g)=\frac{1}{|S|}\sum_{s\in S}f(gs).\]

Question 5.7 If \(G=\mathbf{Z}\), what are the harmonic functions for \(S=\{\pm1\}\)?

Question 5.8 Prove that a harmonic function that has a maximum or a minimum is constant.

Question 5.9 Prove that a function \(f\colon G\to \mathbf{R}\) is Lipschitz (for the word distance on \(G\) associated to \(S\)) if there \(C>0\) such that for all \(g\in G\) and \(s\in S\)

\[|f(gs)-f(g)|\leq C.\]

A crucial in the proof of Gromov’s theorem is the following difficult result due to Kleiner. We will assume it.

Theorem 5.4 (Kleiner theorem) If \(G\) has polynomial growth then the space of Lipschitz harmonic functions \(\mathrm{HL}_S(G)\) has finite dimension and contains at least one non-constant function.

For a Lipschitz function \(f\), we define its Lipschitz norm \(||F||_{\mathrm{Lip}}\) to be the least \(C>0\) such that

\[|f(g)-f(h)|<Cd_S(g,h).\]

Question 5.10

Prove that the Lipschitz norm is a semi-norm on \(\mathrm{HL}_S(G)\) and that \(G\) acts by precomposition on \(\mathrm{HL}_S(G)\) preserving this seminorm.
What are the functions \(f\in\mathrm{HL}_S(G)\) such that \(||f||_{\mathrm{Lip}}=0\)? We denote \(N\) this space.
Prove that the Lipschitz norm induces a norm on \(E=\mathrm{HL}_S(G)/N\) on \(G\) acts by linear isometries on this space
Prove that there is a \(G\)-invariant scalar product on \(E\).

5.8 The compact linear case

In this section, we consider a subgroup \(G\) of \(\mathrm{U}_n(\mathbf{C})\). For \(g\in\mathrm{U}_n(\mathbf{C})\), we denote by \(||g||\) its operator norm.

Question 5.11 Prove that for \(g,h\in \mathrm{U}_n(\mathbf{C})\),

\[||[g,h]-1||\leq 2||g-1||||h-1||.\]

Our aim in this section is to prove the following theorem.

Theorem 5.5 Let \(G\) be a finitely generated subgroup of \(\mathrm{U}_n(\mathbf{C})\) of polynomial growth then \(G\) is virtually abelian.

Let \(G\) fix as in the theorem. For \(\varepsilon\in(0,1/10)\), define \(G'\) as the subgroup of \(G\) defined by \(g\in G\) such that \(||g-I_n||<\varepsilon\) (where \(||\ ||\) is the operator norm).

Question 5.12 Prove there is a constant \(C\) depending on \(n\) and \(\varepsilon\) such that \(G'\) has index at most \(C\) in \(G\).

Question 5.13 Assume \(G'\) has a central element \(g\) that is not a homothety (i.e. not \(\lambda\mathrm{Id}\) for some \(\lambda\in\mathbf{C}\)).

Using the spectral theorem prove that the centralizer of \(g\) can be identified with \(\mathrm{U}_{n_1}(\mathbf{C})\times\dots\times\mathrm{U}_{n_k}(\mathbf{C})\) where \(n=n_1+\dots+n_k\).
Prove the theorem under this assumption and by assuming that the theorem is true for lower dimensions.

From now on, we assume that element in the center of \(G'\) are homotheties.

Question 5.14 Prove that \(G'\) is generated by a finite set \(S\) of elements \(g\) such that \(||g-I_n||<\varepsilon\).

Let \(h_1\in S\) that is not a homothety and with \(||h_1-I_n||=\delta_1<\varepsilon\).

Question 5.15 Prove that for any \(g\in S\), \(||[g,h_1]-I_n||<2\delta_1\varepsilon\) and \(\det([g,h_1])=1\).

Question 5.16 Prove that if \([g,h_1]\) is a homothety for all \(g\in S\) then \(h_1\) is central in \(G'\) and we have a contradiction.

Question 5.17 Construct by induction a sequence \((h_k)\) of elements of \(G'\) such that \(h_k=[g_{k-1},h_{k_1}]\) for some \(g_{k-1}\), \(h_k\) is not a homothety and \(||h_k-I_n||=\delta_n<2\varepsilon\delta_{n-1}\).

Question 5.18

Prove that there is \(c>0\) such that all products \(h_1^{i_1}\dots h_m^{i_m}\) are distinct for \(i_1,\dots,i_m\in(0,c/\varepsilon)\).
Prove that there is \(L>0\) such that all products \(h_1^{i_1}\dots h_m^{i_m}\) for \(i_1,\dots,i_m\in(0,c/\varepsilon)\) lie in some ball of radius \(Cm2^m/\varepsilon\) for the word metric associated to \(S\).
Prove this is a contradiction with polynomial growth.

Question 5.19 Gather all what has been proved in this section to prove Theorem 5.5.

5.9 Group finite with infinite abelian quotient

Definition 5.5 Let \(G\) be a finitely generated group. It is said to have polynomial growth of exponent at most \(d\in \mathbf{N}\) if there is \(C>0\) such that the ball of size \(n\) in \(G\) for some finite generating has at most \(Cn^d\) elements.

In this section, we prove the following theorem.

Theorem 5.6 (Induction step for groups with an infinite cyclic quotient) Let \(d\in\mathbf{N}\). Suppose that any finitely generated group of polynomial growth of exponent at most \(d-1\) is virtually nilpotent. Let \(G\) be a finitely generated of polynomial growth of exponent at most \(d\). Assume that \(G'\) is a finite index subgroup of \(G\) with a normal subgroup \(N\leq G'\) such that \(G'/N\) is infinite cyclic then \(G\) is virtually nilpotent.

Question 5.20 Let \(G'\) be a finitely generated group with surjective homomorphism \(\phi\colon G'\to\mathbf{Z}\). Let \(\{s_1,\dots,s_m\}\) a finite generating set. Use Bézout’s lemma to prove that we can find a generating set \(\{e_1,\dots,e_m\}\) such that \(\{e_1,\dots,e_{m-1}\}\) and their conjugates \(e^k_me_ie^{-k}_m\) by powers of \(e_m\) generates \(\ker\phi\).

Question 5.21 Let \(S_k=\{e^{k'}_me_ie^{-k'}_m,\ |k'|\leq k,\ 1\leq i\leq m-1\}\) and \(B_k\) be the elements in the subgroup generated by \(S_k\) at length at most \(k\) (for the word length given by \(S_k\)).

Prove that if \(S_{k+1}\nsubseteq B_kB^{-1}_k\) then \(|B_{k+1}|\geq 2|B_k|\).
Deduce that for some \(k\) large enough \(S_{k+1}\subseteq B_kB^{-1}_k\).
Prove that for some \(k\) large enough \(S_k\) generates \(\ker\phi\).

We fix \(k\) such that \(S_k\) generates \(\ker\phi\).

Question 5.22 Prove that the ball of radius \(R\) generated by \(S_k\) and \(e_m\) is at least \(R/2\) times as large as the ball of radius \(R/2\) generated by \(S_k\).

Prove that \(\ker\phi\) is finitely generated with polynomial growth at most \(d-1\).

Conclude that \(\ker\phi\) is virtually nilpotent.

Question 5.23 Prove that \(\ker\phi\) contains a normal finite index nilpotent subgroup \(N\) and there is some power \(M\) such that for any element \(g\in\ker\phi\), \(g^M\in N\).

Let \(N'\) be the subgroup of \(\ker\phi\) generated by these powers \(g^M\). Prove this is a nilpotent normal subgroup of \(\ker\phi\).

Question 5.24 Let \(M\in\mathbf{N}\). Let \(Q\) be a finitely generated group that is nilpotent and such that any element satisfies \(g^M=1\). Prove that \(Q\) is finite and deduce that \(N'\) has finite index in \(N\) and thus has finite index in \(\ker\phi\).

Question 5.25 The (infinite) cyclic group generated by \(e_m\) acts by on \(\ker\phi\) by conjugations.

Prove that \(N'\) is invariant for this action and we can construct the semi-direct product \(G''=\mathbf{Z}\ltimes_{e_m}N'\)
Prove that \(G''\) has finite index in \(G\) and thus has polynomial growth.

Let \(G\) be a group, \(g\in G\) and \(H\) a subgroup of \(G\) normalized by \(g\). We say that \(g\) acts unipotently on \(H\) if there \(k\in \mathbf{N}\) such for any iterated commutators \([x_1,[x_2,[\dots[x_{n-1},x_{n}]]]]\) is trivial when \(x_i=g\) or \(x_i\in H\) and \(|\{i,\ x_i=g\}|\geq k\).

Question 5.26 Prove that if there is some power \(e_m^a\) that acts unipotently on \(N'\) then \(G''\) is virtually nilpotent.

Question 5.27 Let \(Z(N')\) be the center of \(N'\).

Prove that \(Z(N')\) is a finitely generated abelian group.
Assume that \(N'\) is nilpotent in \(m\) steps. Prove that \(N'/Z(N')\) is nilpotent in \(m-1\) steps.
Prove there is a finite abelian group \(H\) and some \(l\) such that \(Z(N')\simeq \mathbf{Z}^l\times H\).
Prove that \(H\) is a characteristic subgroup (invariant under automorphisms) of \(Z(N')\).
Prove that any automorphism of \(Z(N')\) induces an automorphism of \(H\).
Prove that there \(a\in\mathbf{N}\) large enough such that the action induced by conjugation by \(e_m^a\) is trivial on \(H\).
Prove that \(e_m^a\) induces an action on \(\mathbf{Z}^l\simeq Z(N')/H\).

Question 5.28 Prove that it suffices (arguing by induction on the number of steps of nilpotency) to prove there is some power \(e_m^a\) that acts unipotently on \(Z(N')\).

Question 5.29 Prove that the group of automorphisms of \(\mathbf{Z}^l\) is \(\mathrm{SL}_l(\mathbf{Z})\).

Question 5.30 Let \(A\in \mathrm{SL}_l(\mathbf{Z})\) be the matrix corresponding to the action of \(e_m^a\) on \(\mathbf{Z}^l\).

Prove that \(||A^n||\) cannot grow exponentially and thus \(A\) has unit complex eigenvalues.
By Kronecker theorem about root of unity, all eigenvalues of \(A\) are roots of unity
Prove that some power of \(A\) is unipotent (i.e. there is some power \(k\) such that \((A-I_l)^k=0\).)
Prove that the action of some power of \(e_m^a\) on \(Z(N')\) is unipotent.

Question 5.31 Explain why all what has been done in this section prove Theorem 5.6

5.10 Putting all together

Let \(G\) be a finitely generated group with polynomial growth with of exponent at most \(d\).

We prove by induction on \(d\) that \(G\) is virtually nilpotent.

Question 5.32 What is the base case? Prove it.

From now on we assume Gromov’s theorem for exponent less than some fix \(d\).

Question 5.33 Let us consider the action of \(G\) on the quotient space \(E\) from Section 5.7. Prove that the image \(A\) of \(G\) in \(\mathrm{GL}(E)\) is virtually abelian.

Question 5.34
1. Prove that if \(A\) is infinite then \(G\) has a finite index subgroup with a homomorphism to \(\mathbf{Z}\) with infinite image. 2. Using Theorem 5.6, prove we are done.

Question 5.35
1. Prove that if \(A\) is finite image than \(G\) has a finite index subgroup \(G'\) that acts trivially on \(E\). 2. Prove in that case, that for any \(g\in G'\) and \(f\in \mathrm{HL}(G)\), \(g\cdot f=f+\lambda_g(f)\) where \(\lambda_g(f)\in\mathbf{R}\). 3. Prove that for any \(g\in G'\), \(\lambda_g\) is a linear functional on \(\mathrm{HL}(G)\) that yields a homomorphism \(G'\to(\mathrm{HL}(G)*,+)\).

Question 5.36 Prove that if \(g\mapsto\lambda_g\) has infinite image, we can use Theorem 5.6 to conclude.

Question 5.37
1. Prove that if \(g\mapsto\lambda_g\) has finite image, then there is another finite index subgroup \(G''\) that acts trivially on \(\mathrm{HL}(G)\). 2. Prove in that case that all Lipschitz harmonic functions take only finitely values. 3. Prove that this implies that all harmonic Lipshcitz harmonic functions are constant and this is a contradiction.