## Notes of Enrico Le Donne’s Oxford lecture 24-03-2017

From homogeneous spaces to Lie groups

With Michael Cowling, Ville Kivioja, Alessandro Ottazi and Sebastiano Nicolussi Golo.

1. Homogeneous metric spaces

I.e. with a transitive group of isometries.

Solenoids (inverse limits of ${S^1}$ with maps ${t\mapsto 2t}$) are homogeneous. ${p}$-adics are too. We shall stick to connected metric spaces.

From the solution to Hilbert’s 5th problem, it follows that a connected, locally connected and locally compact homogeneous space is ${G/H}$ with ${G}$ a Lie group and ${H}$ a compact group. Call this a Lie homogeneous space.

It follows that if ${X}$ is a connected and locally compact homogeneous space, then for all ${\epsilon>0}$, ${X}$ is ${(1,\epsilon)}$-quasi-isometric to a metric Lie group (choose the metric on the isometry group ${G}$ in order that ${H}$ has small diameter). Up to passing to a subgroup, one can assume ${G}$ to be solvable.

2. Isometric metric Lie groups

Theorem 1 Given a metric Lie group ${(G,d_G)}$, there is a ${(1,\epsilon)}$-qi invariant metric ${d'_G}$ which is isometric to a metric Lie group of the form ${K\times S}$ where ${K}$ is compact and ${S}$ is solvable.

Example. ${\widetilde{Sl_2}}$ can be made isometric to ${\mathbb{H}^2 \times{\mathbb R}}$.

3. The polynomial growth case

For a general (non-geodesic) metric Lie group, polynomial growth does not imply doubling. Nevertheless,

Theorem 2 For a metric Lie group ${(G,d_G)}$ of polynomial growth, there is a ${(1,\epsilon)}$-qi invariant metric ${d'_G}$ which is isometric to a metric Lie group of the form ${K\times N}$ where ${K}$ is compact and ${N}$ is nilpotent.

${N}$ is uniquely defined, it is the nilshadow of the solvable group ${S}$.

In fact, the following can be extracted from work of Gordon and Wilson. Given simply connected Lie groups ${H}$ and ${N}$, with ${N}$ nilpotent, then ${H}$ and ${N}$ can be made isometric iff ${H}$ is solvable of type (R) and ${N}$ is its nilshadow.

4. Playing with distances

On ${{\mathbb R}^n}$, there is a wealth of left-invariant distances (snowfloakes, bounded,…). Idem for nilpotent groups. Geodesicity, existence of one dilation suffices to characterize Carnot groups with sub-Finsler metrics.

What if geodesicity is relaxed?

We consider gradings on Lie algebras indexed by real numbers ${\geq 1}$. Such algebras are automatically nilpotent.

Theorem 3 (Hebisch-Sikora) Every simply connected Lie group whose Lie algebra is graded admits an invariant distance which is dilation invariant.

Example. Heisenberg with 3 homogeneous components.

Theorem 4 (Hebisch-Sikora) If ${X}$ is a metric space which is locally compact, connected homogeneous and admit one dilation. Then ${X}$ is a graded Lie group with a left-invariant metric and homogeneous under dilations.

This a combination of many existing results. Since ${X}$ is doubling, ${ISo(X)}$ is Lie with polynomial growth. The Killing orthogonal to a stabilizer exponentiates into a subgroup with a simply transitive action on ${X}$. So ${X}$ is a metric Lie group.

5. Quiz

Say finitely generated groups are equivalent if they admit isometric word metrics. Is this an equivalence relation?