Notes of Yves Cornulier’s Rennes lecture nr 2

During the first lecture, we saw that, up to compact and cocompact groups, locally compact groups of polynomial growth can be reduced to simply connected Lie groups. Therefore we continue with a thorough study of these groups.

1. The lower central series

${G^{(1)}=G}$, ${G^{(i+1)}=[G,G^{(i)}]}$. For simply connected nilpotent Lie groups, the Lie algebra functor is an equivalence of categories. In fact, thanks to the Baker-Campbell-Hausdorff formula, the group can be viewed as a multiplication on the Lie algebra.

Pick a complement ${V_i}$: ${\mathfrak{g}^{(i)}=V_i \oplus\mathfrak{g}^{(i+1)}}$. Fix a Euclidean norm on each ${V_i}$, denote by ${V_i(R)}$ the ${R}$-ball in ${V_i}$. Get a Euclidean norm on ${\mathfrak{g}}$ and a left-invariant Riemannian metric on ${G}$.

Guivarc’h showed that that ${r}$-ball in ${G}$ is squeezed between boxes

$\displaystyle \begin{array}{rcl} K(R)=\bigoplus_i V_i(R^i) \end{array}$

of respective radii ${R=r/C}$ and ${R'=Cr}$. It follows that growth is polynomial of degree

$\displaystyle \begin{array}{rcl} \delta=\sum_{i}i\,\mathrm{dim}(V_i). \end{array}$

Examples

1. The standard filiform Lie algebra ${\mathfrak{f}_n}$ has a basis ${x_1,y_2,\ldots,y_n}$ with only nonzero brackets ${[x_1,y_i]=y_{i+1}}$ for ${i=2,\ldots,n-1}$. The Lie group ${F_n}$ is a semi-direct product ${{\mathbb R}\times{\mathbb R}[t]/(t^{n-1})}$, where the generator ${s}$ of ${{\mathbb R}}$ acts by ${sP(t)=(1+t)^s P(t)}$. One can take ${V_1=}$ and ${V_i=}$ for ${i\geq 2}$, ${\delta=1+\frac{n(n-1)}{2}}$.
2. The Heisenberg Lie algebra ${H_{2n+1}}$ has a basis ${x_1,\ldots,x_n,y_1,\ldots,y_n,z}$ with nonzero brackets ${[x_i,y_i]=z}$. ${V_1=}$, ${V_2=}$, ${\delta=2n+2}$.
3. Upper unipotent matrices. Here ${\mathrm{dim}(V_i)=\max(0,n-i)}$.
4. Free ${s}$-nilpotent Lie group on ${k}$ generators.
5. Polynomial vectorfields on the line. Generators are ${e_i=x^{i+1}\frac{\partial}{\partial x}}$, nonzero brackets are ${[e_i,e_j]=(i-j)e_{i+j}}$.

2. Carnot Lie algebras

Definition 1 A nilpotent Lie algebra ${\mathfrak{g}}$ is Carnot if it satisfies one of the equivalent properties

1. ${\mathfrak{g}}$ admits a Lie algebra grading ${\mathfrak{g}=\bigoplus_i \mathfrak{g}_i}$ such that ${\mathfrak{g}^{(i)}=\bigoplus_{j\geq i}g_i}$.
2. ${\mathfrak{g}}$ has a contracting automorphism inducing a homothety on ${\mathfrak{g}/\mathfrak{g}^{(2)}}$.
3. ${\mathfrak{g}}$ has a self-derivation inducing a identity on ${\mathfrak{g}/\mathfrak{g}^{(2)}}$.
4. ${\mathfrak{g}}$ is isomorphic to ${Car(\mathfrak{g})}$.
5. The corresponding Lie group admits a proper, geodesic, left-invariant distance with non-isometric similarities.

Here ${Car(\mathfrak{g})}$, the associated Carnot algebra, is the natural Lie algebra structure on ${\bigoplus_{i}\mathfrak{g}^{(i)}/\mathfrak{g}^{(i+1)}}$. It can be thought of as a first order approximation of ${\mathfrak{g}}$.

In the above list, all examples are Carnot but the last one, polynomial vectorfields. The associated Carnot algebra is filiform.

3. Quasiisometry classification

Question. Does quasiisometry imply isomorphism ?

Pansu 1989 : quasiisometric nilpotent Lie groups have isomorphic associated Carnot Lie algebras.

Shalom 2001 : quasiisometric nilpotent Lie groups have the same Betti numbers.

4. Questions

Expansion of volume growth ? Breuillard gave an upper bound of ${r^{\delta-2/3s}}$ on the second term.