## Notes of Yves Cornulier’s Rennes lecture nr 1

Geometry of nilpotent groups and groups of polynomial growth

1. Growth

Definition. Let ${G}$ be a locally compact group equipped with a left Haar measure ${\lambda}$. Assume ${G}$ has a compact generating set ${S}$. The growth function is ${b_S(n)=\lambda(S^n)}$.

Its asymptotic behaviour is independant on ${S}$, where we write, for functions ${{\mathbb R}\rightarrow{\mathbb R}}$, ${f\leq g}$ if ${f(n)\leq Cg(C'n)+C''}$, and ${f\sim g}$ if ${f\leq g\leq f}$.

The study of growth began in the 60’s with Milnor, Svarc, especially for discrete or connected Lie groups.

Examples. 1. ${{\mathbb R}^d}$ has polynomial growth of degree ${d}$.

2. The affine group of the line is a semi-direct product ${{\mathbb R}\times{\mathbb R}}$. It is the set of matrices ${\begin{pmatrix} 1 & x \\ 0 & e^t \end{pmatrix}}$ where ${x}$, ${t\in{\mathbb R}}$. ${\lambda=dx\,dt}$. The ${n}$-ball contains ${\{(x,t)\,;\,-e^n \leq x\leq e^n,\,-n\leq t\leq n\}}$, so growth is exponential.

3. The Heisenberg group is the set of matrices ${\begin{pmatrix} 1 & x &z\\ 0 & 1&y\\ 0&0&1 \end{pmatrix}}$ where ${x}$, ${y}$, ${z\in{\mathbb R}}$. ${\lambda=dx\,dy\,dz}$. The ${n}$-ball contains ${\{(x,y,z)\,;\,|x|\leq n,\,|y|\leq n,\,|z|\leq n^2,\}}$, so growth is polynomial of degree 4.

2. The case of Lie groups

These 3 examples are representative of the general case for Lie groups.

Theorem 1 (Guivarc’h, 1973, then a PhD student in Rennes) Let ${G}$ be a virtually connected Lie group. Then ${G}$ has polynomial growth if and only if all eigenvalues of the adjoint representation of ${G}$ on ${\mathfrak{g}}$ belong to the unit circle. Otherwise, ${G}$ has exponential growth.

Remark. Nilpotent Lie groups correspond to the case where 1 is the only eigenvalue. The isometry group of ${{\mathbb R}^d}$ is an example where eigenvalues can be non real.

3. Gromov’s polynomial growth theorem

Theorem 2 (Gromov 1981) Let ${G}$ be a discrete, finitely generated group with polynomial growth. Then ${G}$ is virtually nilpotent (and conversely).

Question. Does nonpolynomial growth imply exponential growth ?

Theorem 3 (Tits 1972, Grigorchuk 1984) Answer is yes for linear groups.

Answer is no for general discrete groups.

Open for finitely presented groups.

4. Generalization to locally compact groups

Theorem 4 (Losert 1987) Let ${G}$ be a locally compact, compactly generated group with polynomial growth. Then ${G}$ is compact-by-Lie, i.e. is has a compact normal subgroup ${W}$ such that ${G/W}$ is Lie. (Here Lie, means Lie, i.e. connected Lie-by-discrete.)

Let ${G}$ be a compactly generated Lie group. Then ${G}$ has polynomial growth if and only if

1. ${G^0}$ has polynomial growth,
2. the discrete quotient ${G/G^0}$ is virually nilpotent.
3. the action of ${G}$ on ${\mathfrak{g}}$ has its eigenvalues on the unit circle.

5. Malcev completion

Theorem 5 Let ${\Gamma}$ be a finitely generated nilpotent group. Then there is a homomorphism ${\Gamma\rightarrow\Gamma_{\mathbb Q}\rightarrow\Gamma_{\mathbb R}}$ where

1. ${i:\Gamma\rightarrow\Gamma_{\mathbb Q}}$ has finite kernel (the set of torsion elements of ${\Gamma}$),
2. ${\Gamma_{\mathbb Q}}$ is isomorphic to the set of rational points of an unipotent algebraic group,
3. ${\Gamma_{\mathbb Q}}$ is nilpotent, uniquely divisible (${x\mapsto x^n}$ is injective for all ${n}$), the image ${i(\Gamma)}$ intersects every nontrivial subgroup of ${\Gamma_{\mathbb Q}}$,
4. ${i(\Gamma)}$ is a cocompact lattice in ${\Gamma_{\mathbb R}}$,
5. the sequence is functorial.

Definition 6 Say that a locally compact group of polynomial growth is universal if it is a semidirect product ${U\times K}$ where ${U}$ is a simply connected nilpotent Lie group and ${K}$ is a maximal compact group of automorphisms of ${U}$. ${U}$ is called the nilshadow of ${G}$.

Proposition 7 (Essentially due to Breuillard) Every locally compact group of polynomial growth ${G}$ admits an essentially unique homomorphism to a universal one, with compact kernel and cocompact image.