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 1 Given 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 2 For 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.
Say finitely generated groups are equivalent if they admit isometric word metrics. Is this an equivalence relation?