** 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 with maps ) are homogeneous. -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 with a Lie group and a compact group. Call this a Lie homogeneous space.

It follows that if is a connected and locally compact homogeneous space, then for all , is -quasi-isometric to a metric Lie group (choose the metric on the isometry group in order that has small diameter). Up to passing to a subgroup, one can assume to be solvable.

**2. Isometric metric Lie groups **

Theorem 1Given a metric Lie group , there is a -qi invariant metric which is isometric to a metric Lie group of the form where is compact and is solvable.

**Example**. can be made isometric to .

**3. The polynomial growth case **

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

Theorem 2For a metric Lie group of polynomial growth, there is a -qi invariant metric which is isometric to a metric Lie group of the form where is compact and is nilpotent.

is uniquely defined, it is the nilshadow of the solvable group .

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

**4. Playing with distances **

On , 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 . 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 is a metric space which is locally compact, connected homogeneous and admit one dilation. Then is a graded Lie group with a left-invariant metric and homogeneous under dilations.

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

**5. Quiz **

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