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?

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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