On the classification of Lie groups by metric geometryusers.jyu.fi/~vikakivi/prujut/2018-11-13.pdf · Preliminaries MetricgeometryonLiegroups MetricclassiﬁcationofLiegroups OntheclassiﬁcationofLiegroups

PreliminariesMetric geometry on Lie groups

Metric classification of Lie groups

On the classification of Lie groupsby metric geometry

Ville Kivioja

University of Jyväskylä, Finland, EU

Seminar on Groups and Geometry, 13 November 2018In UNSW, Sydney, Australia

These slides are available at my home pagehttp://users.jyu.fi/~vikakivi/

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http://users.jyu.fi/~vikakivi/



1 Preliminaries

2 Metric geometry on Lie groups

3 Metric classification of Lie groups

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1 Preliminaries



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Lie group

The pair (G , ∗) is a Lie group if G is a differentiable manifold andthe map

∗ : G × G → G (g , h) 7→ g ∗ h ≡ gh

is a smooth group operation with smooth inversion.

Lie algebra

A real vector space V equipped with a bilinear operation[·, ·] : V × V → V is a Lie algebra if ∀X ,Y ,Z ∈ V

[X ,Y ] = −[Y ,X ] (antisymmetry)[X , [Y ,Z ]] + [Y , [Z ,X ]] + [Z , [X ,Y ]] = 0 (Jacobi identity)

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Lie algebra



Every Lie group has an associated Lie algebra. This Lie algebra isthe vector space of left-invariant vector fields on the manifoldequipped with the operation [X ,Y ] = XY − YX (recall that vectorfields are differential operators).

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Lie algebra




G Lie group −→ Lie(G ) .= g Lie algebra

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Lie algebra




G simply connectedLie group ←→ gfinite dimLie algebra

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Lie algebra




G simply connectedLie groupLie correspondence←→ g finite dimLie algebra

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Classification of Lie groups means to answer the following question:What are the different types of Lie groups there are?

One is supposed to come up with a ‘concrete’ list of groups, forexample dimension by dimension, so that given a Lie group, one ofits isomorphic copies can by found from the list.

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What are the different types of Lie groups of dim n?

A general strategy: Reduce into two questions:What are the simply connected Lie groups of dimension n?⇔ What are the Lie algebras of dimension n.What are the discrete normal subgroups of the simplyconnected groups (the universal covers)?

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A general strategy: Reduce into two questions:

What are the simply connected Lie groups of dimension n?⇔ What are the Lie algebras of dimension n.What are the discrete normal subgroups of the simplyconnected groups (the universal covers)?

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A general strategy: Reduce into two questions:What are the simply connected Lie groups of dimension n?

⇔ What are the Lie algebras of dimension n.What are the discrete normal subgroups of the simplyconnected groups (the universal covers)?

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A general strategy: Reduce into two questions:What are the simply connected Lie groups of dimension n?⇔ What are the Lie algebras of dimension n.

What are the discrete normal subgroups of the simplyconnected groups (the universal covers)?

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A general strategy: Reduce into two questions:What are the simply connected Lie groups of dimension n?⇔ What are the Lie algebras of dimension n.What are the discrete normal subgroups of the simplyconnected groups (the universal covers)?

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Classification of simply connected Lie groups of dimension 3.

Done in 1897 by Luigi Bianchi:

R3

Heis

S̃L(2)

S̃E(2)

SOL (i.e. D−1)

Aff(R)◦ × R

D1

S3

J

{Cλ | λ > 0}

{Dλ | λ ∈ ]0, 1[ }

{Dλ | λ ∈ ]−1, 0[ }

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1 Preliminaries



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Metric geometry on Lie groups

Whenever one wants to consider a Lie group as a metric space, onemust assume a distance function that

induces the manifold topologyis left-invariant: d(p, q) = d(gp, gq) ∀p, q, g .

Such a distance function will be called compatible.

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An example: The half-plane R× R+ with the operation

(a, t) ∗ (b, s) = (tb + a, ts)

is a 2D Lie group.

The hyperbolic metric of the half-plane is compatible, since for anyfixed (a, t) ∈ R× R+ the left-translation map

(b, s) 7→ (tb + a, ts)

is an isometry of the hyperbolic half-plane. Consider for examplethe cases a = 0 (homotheties) or t = 1 (horisontal translations).

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A Lie group G with a compatible distance function d will be calledmetric Lie group (G , d).

differentiable manifold group

Lie group metric space

metric Lie group

Riemannian Lie groups form an important and quite well studiedclass of examples (see e.g. Milnor: Curvatures of left-invariantmetrics on Lie groups)

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A Lie group does not have a canonical (compatible) distance.

Compatible distances can always be constructed:An example is to choose an inner product on tangent space of theneutral element, and extend this to Riemannian metric byleft-translations.

BUTTwo left-invariant Riemannian distances are not necessarilyisometric, not even conformal (easy examples in Heis)In the class of compatible distances there are alsosubRiemannian distances, and non-geodesic distances likecut-of-distances (e.g. min{1, dE}) and snowflakes etc.

So two compatible distances can be arbitrarily different, no metricrelation between them. . .

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Compatible distances can always be constructed:

An example is to choose an inner product on tangent space of theneutral element, and extend this to Riemannian metric byleft-translations.



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BUT

Two left-invariant Riemannian distances are not necessarilyisometric, not even conformal (easy examples in Heis)In the class of compatible distances there are alsosubRiemannian distances, and non-geodesic distances likecut-of-distances (e.g. min{1, dE}) and snowflakes etc.


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BUTTwo left-invariant Riemannian distances are not necessarilyisometric, not even conformal (easy examples in Heis)

In the class of compatible distances there are alsosubRiemannian distances, and non-geodesic distances likecut-of-distances (e.g. min{1, dE}) and snowflakes etc.


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However, if two compatible distance functions on a same Lie groupare quasi-geodesic, the distances are quasi-isometric via theidentity map.

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A map f : M → N between two metric spaces is a (L,C )-quasi-isometry with L > 0,C ≥ 0, if

1Ld(x , x ′)− C ≤ d(f (x)), f (x ′)) ≤ Ld(x , x ′) + C ∀x , x ′ ∈ M

and for all y ∈ N there is x ∈ M with d(f (x), y) ≤ C .(L, 0)-quasi-isometry is an L-biLipschitz homeomorphism.(1, 0)-quasi-isometry is an isometry.

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quasi-isometric to a geodesic metric space ⇒ quasi-geodesic

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→ There is a canonical equivalence class of metrics up toquasi-isometry.

→ We may study Lie groups as large-scale geometric objects.

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Studying finitely generated groups as large-scale geometricobjects is actually a program going back to Gromov.

Finitely generated groups are 0-dimensional disconnected (discrete)Lie groups, but the same program of Gromov is nowadays studiedon the world of Lie groups, and more generally locally compactcompactly generated groups (Cornulier, de la Harpe, . . . )

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The quasi-isometric classification problem of Lie groups:Given two Lie groups, decide if they are quasi-isometric or not?

A very general and hard question. . .

When are two different Lie groups of dimension 3 quasi-isometric?

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Classification of simply connected Lie groups of dimension 3:

R3

Heis

S̃L(2)

S̃E(2)

SOL

Aff(R)◦ × R

D1

S3

J

{Cλ | λ > 0}

{Dλ | λ ∈ ]0, 1[ }

{Dλ | λ ∈ ]−1, 0[ }

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Quasi-isometry classes of dimension 3:

R3

Heis

S̃L(2)

S̃E(2)

SOL

Aff(R)◦ × R

D1

S3

J

{Cλ | λ > 0}

{Dλ | λ ∈ ]0, 1[ }

{Dλ | λ ∈ ]−1, 0[ }

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A much finer geometric relation:

We say two Lie groups G and H can be made isometric, writeG ∼ H, if there are compatible distance functions dG on G and dHon H such that

(G , dG )isome∼= (H, dH)

Notice that such a relation is not a priori transitive: If G ∼ H andH ∼ N, this means that there are distance functions dG , dH , d ′H , dNin such a way that

(G , dG )isome∼= (H, dH) and (H, d ′H)

isome∼= (N, dN)

It might still be that (H, dH) is not isometric to (H, d ′H), and henceit might be G 6∼ N. Indeed this situation occurs at least for somenot simply connected groups!

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Notice that such a relation is not a priori transitive:

If G ∼ H andH ∼ N, this means that there are distance functions dG , dH , d ′H , dNin such a way that


isome∼= (N, dN)


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isome∼= (N, dN)


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isome∼= (N, dN)

It might still be that (H, dH) is not isometric to (H, d ′H), and henceit might be G 6∼ N.

Indeed this situation occurs at least for somenot simply connected groups!

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isome∼= (N, dN)


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Theorem (K., Le Donne)

If two Lie groups can be made isometric, they can be madeisometric using Riemannian distances.

(If time: a few words about the proof later.)

−→ G ∼ H if and only if there is a Riemannian manifold on whichboth G and H act simply transitively (relation to Gordon&Wilson:Isometry groups of Riemannian solvmanifolds)

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Example. The Lie group SE(2) is the group of rigid motions(rotations and translations) of the plane.

Its universal cover S̃E(2)is topologically R3. The Euclidean metric of R3 is left-invariant forthe group structure of S̃E(2):

L(a,b,α)(x , y , θ) = (Rot(α)(x , y) + (a, b), θ + α)

→ S̃E(2) and R3 can be made isometric.

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Example. The Lie group SE(2) is the group of rigid motions(rotations and translations) of the plane. Its universal cover S̃E(2)is topologically R3.

The Euclidean metric of R3 is left-invariant forthe group structure of S̃E(2):



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Example. The Lie group SE(2) is the group of rigid motions(rotations and translations) of the plane. Its universal cover S̃E(2)is topologically R3. The Euclidean metric of R3 is left-invariant forthe group structure of S̃E(2):



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PropositionIf two Lie groups can be made isometric, they are quasi-isometric.

Proof.They can be made isometric using Riemannian distances, and thoseare in the canonical class. This means two distances in therespective canonical classes are isometric. Hence all the distances inthe respective canonical classes are quasi-isometric.

→ Inverse question: Which of the pairs of Lie groups that arequasi-isometric, can indeed be made isometric?

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Proof.

They can be made isometric using Riemannian distances, and thoseare in the canonical class. This means two distances in therespective canonical classes are isometric. Hence all the distances inthe respective canonical classes are quasi-isometric.


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Proof.They can be made isometric using Riemannian distances, and thoseare in the canonical class.

This means two distances in therespective canonical classes are isometric. Hence all the distances inthe respective canonical classes are quasi-isometric.


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Proof.They can be made isometric using Riemannian distances, and thoseare in the canonical class. This means two distances in therespective canonical classes are isometric.

Hence all the distances inthe respective canonical classes are quasi-isometric.


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Proof.They can be made isometric using Riemannian distances, and thoseare in the canonical class. This means two distances in therespective canonical classes are isometric. Hence all the distances inthe respective canonical classes are quasi-isometric.


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1 Preliminaries



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Question: Which of the pairs of Lie groups that arequasi-isometric, can indeed be made isometric?

R3

Heis

S̃L(2)

S̃E(2)

SOL

Aff(R)◦ × R

D1

S3

J

{Cλ | λ > 0}

{Dλ | λ ∈ ]0, 1[ }

{Dλ | λ ∈ ]−1, 0[ }

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Question: Which of the pairs of Lie groups that arequasi-isometric, can indeed be made isometric?

R3

Heis

S̃L(2)

S̃E(2)

SOL

Aff(R)◦ × R

D1

S3

J

{Cλ | λ > 0}

{Dλ | λ ∈ ]0, 1[ }

{Dλ | λ ∈ ]−1, 0[ }

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A general metric classification problem: To completely answerto the following: Given two different Lie groups,

are they quasi-isometric?IF YES, can they be made isometric?

ConjectureIf two nilpotent Lie groups are quasi-isometric, they must beisomorphic.

TheoremIf two nilpotent Lie groups can be made isometric, they must beisomorphic.

Dimension 3 is solved.→ Can we answer these questions in dimension 4? or 5?

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are they quasi-isometric?

IF YES, can they be made isometric?




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are they quasi-isometric?IF YES, can they be made isometric?




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General strategy, top-down

Given two Lie groups, how to conclude that they are notquasi-isometric?

One usually proves that some QI-invariant differ on them.i) possible polynomial growth and its degree.ii) possible Gromov-hyperbolicity and the topology of the

resulting visual boundary.iii) For solvable groups of the form Rn oR, the real part Jordan

form of the action matrix (Xie).iv) For nilpotent groups, the isomorphism class of the associated

Carnot group (Pansu) and the real cohomology algebra (Sauer)

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One usually proves that some QI-invariant differ on them.

i) possible polynomial growth and its degree.ii) possible Gromov-hyperbolicity and the topology of the




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One usually proves that some QI-invariant differ on them.i) possible polynomial growth and its degree.

ii) possible Gromov-hyperbolicity and the topology of theresulting visual boundary.

iii) For solvable groups of the form Rn oR, the real part Jordanform of the action matrix (Xie).

iv) For nilpotent groups, the isomorphism class of the associatedCarnot group (Pansu) and the real cohomology algebra (Sauer)

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resulting visual boundary.

iii) For solvable groups of the form Rn oR, the real part Jordanform of the action matrix (Xie).


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form of the action matrix (Xie).


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General strategy, bottom-up

Given two Lie groups, how to conclude that they can be madeisometric?

TheoremLet H and N be two simply connected Lie groups, with Nnilpotent. Then the following are equivalent:i) H can be made isometric to N.ii) H is solvable and has polynomial growth, and N is the

nilshadow of H.

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TheoremLet H and N be two simply connected Lie groups, with Nnilpotent. Then the following are equivalent:

i) H can be made isometric to N.ii) H is solvable and has polynomial growth, and N is the

nilshadow of H.

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TheoremLet H and N be two simply connected Lie groups, with Nnilpotent. Then the following are equivalent:i) H can be made isometric to N.

ii) H is solvable and has polynomial growth, and N is thenilshadow of H.

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nilshadow of H.

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nilshadow of H.

Abelian ⊂ nilpotent

{⊂ solvable⊂ polynomial growth ⊂ unimodular

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nilshadow of H.

solvable nilshadow−−−−−→ nilpotent

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nilshadow of H.

Example: The group S̃E(2) has polynomial growth and its nilshadowis R3.

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nilshadow of H.

Consequences:Two different nilpotent groups cannot be made isometric(known earlier).Only groups that can be made isometric to nilpotent groupsare solvable and have polynomial growth.Every solvable group having polynomial growth can be madeisometric to one and only one nilpotent group (nilshadow isunique).

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nilshadow of H.

Consequences:Two different nilpotent groups cannot be made isometric(known earlier).

Only groups that can be made isometric to nilpotent groupsare solvable and have polynomial growth.Every solvable group having polynomial growth can be madeisometric to one and only one nilpotent group (nilshadow isunique).

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nilshadow of H.

Consequences:Two different nilpotent groups cannot be made isometric(known earlier).Only groups that can be made isometric to nilpotent groupsare solvable and have polynomial growth.

Every solvable group having polynomial growth can be madeisometric to one and only one nilpotent group (nilshadow isunique).

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nilshadow of H.

Consequences:Two different nilpotent groups cannot be made isometric(known earlier).Only groups that can be made isometric to nilpotent groupsare solvable and have polynomial growth.Every solvable group having polynomial growth can be madeisometric to one and only one nilpotent group (nilshadow isunique).

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→ Complete metric classification of solvable Lie groups ofpolynomial growth in dimension 4.

R4

S̃E(2) × R

Heis × R

A4,10

Engel

A4,10 : [E2,E3] = E1 [E2,E4] = −E3 [E3,E4] = E2

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R4

S̃E(2) × R

Heis × R

A4,10

Engel

A4,10 : [E2,E3] = E1 [E2,E4] = −E3 [E3,E4] = E2

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R4

S̃E(2) × R

Heis × R

A4,10

Engel

A4,10 : [E2,E3] = E1 [E2,E4] = −E3 [E3,E4] = E2

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nilshadow of H.

Given dimension n:

i) Find all simply connected solvable Lie groups of polynomialgrowth (nilpotent and not).

ii) Calculate the nilshadows of the non-nilpotent groups.

−→ The non-nilpotent groups can be made isometric to itsnilshadow, and not to anything else. The nilpotent groups cannotbe made isometric to other nilpotent groups.

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Polynomial growth, dim 5

As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan34 / 43




As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan35 / 43




As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan36 / 43




As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan37 / 43




As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan38 / 43




As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan

?

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Finally, we may observe some transitivity in this setting.

Theorem (Jablonski -18)

Let G be a solvable unimodular simply connected Lie group. Thenthere is a left-invariant Riemannian metric g such that for everyother left-invariant Riemannian metric g ′ we have

Isome(G , g ′) ⊂ Isome(G , g)

up to conjugation by a diffeomorphism.

Corollary

Assume H,H ′ are simply connected solvable groups of polynomialgrowth that have the same nilshadow N. Then H ∼ H ′.

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As5,17

A4,10 × R

A5,14

A5,18

A±5,26

R5

Heis × R2

Engel × R

A5,2

Heis(5)

A5,1

A5,6

A5,5

Cartan

?

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Theorem (K., Le Donne)

If two Lie groups can be made isometric, they can be madeisometric using Riemannian distances.

Proof.Self-isometries of a metric Lie group are smooth (AA + MZ).Stabilisers of the isometry group are compact.Average a scalar product over a stabiliser and left-translate togreate a Riemannian metric with extended isometry group.Push this metric forward to the latter group: the result is aleft-invariant Riemannian metric (left-invariance needs that Fwas isometry for the original left-invariant metrics).

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Summary

On Lie groups one studies two types of ”metric structure”, onebeing stronger than the other, based on which two groups canbe related or not.

Being quasi-isometric is an equivalence relation. For ‘can bemade isometric’ this could be the case for simply connectedgroups, we don’t know yet.For those dimensions for which the Lie groups are classified, itis natural to try to classify them by the means of metricgeometry.This means finding the quasi-isometry equivalence classes, andthe pairs in the classes that can be made isometric.Complete solution is available ina) dimension 3b) dimension 4 for polynomial growthc) dimension 5 for polynomial growth except one pair

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Summary

On Lie groups one studies two types of ”metric structure”, onebeing stronger than the other, based on which two groups canbe related or not.Being quasi-isometric is an equivalence relation. For ‘can bemade isometric’ this could be the case for simply connectedgroups, we don’t know yet.

For those dimensions for which the Lie groups are classified, itis natural to try to classify them by the means of metricgeometry.This means finding the quasi-isometry equivalence classes, andthe pairs in the classes that can be made isometric.Complete solution is available ina) dimension 3b) dimension 4 for polynomial growthc) dimension 5 for polynomial growth except one pair

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Summary

On Lie groups one studies two types of ”metric structure”, onebeing stronger than the other, based on which two groups canbe related or not.Being quasi-isometric is an equivalence relation. For ‘can bemade isometric’ this could be the case for simply connectedgroups, we don’t know yet.For those dimensions for which the Lie groups are classified, itis natural to try to classify them by the means of metricgeometry.

This means finding the quasi-isometry equivalence classes, andthe pairs in the classes that can be made isometric.Complete solution is available ina) dimension 3b) dimension 4 for polynomial growthc) dimension 5 for polynomial growth except one pair

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Summary

On Lie groups one studies two types of ”metric structure”, onebeing stronger than the other, based on which two groups canbe related or not.Being quasi-isometric is an equivalence relation. For ‘can bemade isometric’ this could be the case for simply connectedgroups, we don’t know yet.For those dimensions for which the Lie groups are classified, itis natural to try to classify them by the means of metricgeometry.This means finding the quasi-isometry equivalence classes, andthe pairs in the classes that can be made isometric.

Complete solution is available ina) dimension 3b) dimension 4 for polynomial growthc) dimension 5 for polynomial growth except one pair

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Summary

On Lie groups one studies two types of ”metric structure”, onebeing stronger than the other, based on which two groups canbe related or not.Being quasi-isometric is an equivalence relation. For ‘can bemade isometric’ this could be the case for simply connectedgroups, we don’t know yet.For those dimensions for which the Lie groups are classified, itis natural to try to classify them by the means of metricgeometry.This means finding the quasi-isometry equivalence classes, andthe pairs in the classes that can be made isometric.Complete solution is available ina) dimension 3

b) dimension 4 for polynomial growthc) dimension 5 for polynomial growth except one pair

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Summary

On Lie groups one studies two types of ”metric structure”, onebeing stronger than the other, based on which two groups canbe related or not.Being quasi-isometric is an equivalence relation. For ‘can bemade isometric’ this could be the case for simply connectedgroups, we don’t know yet.For those dimensions for which the Lie groups are classified, itis natural to try to classify them by the means of metricgeometry.This means finding the quasi-isometry equivalence classes, andthe pairs in the classes that can be made isometric.Complete solution is available ina) dimension 3b) dimension 4 for polynomial growthc) dimension 5 for polynomial growth except one pair

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PreliminariesMetric geometry on Lie groupsMetric classification of Lie groups

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On the classification of Lie groups by metric geometryusers.jyu.fi/~vikakivi/prujut/2018-11-13.pdf · Preliminaries MetricgeometryonLiegroups MetricclassiﬁcationofLiegroups OntheclassiﬁcationofLiegroups