Convex representations and their geodesic flows

Transcription

1 Convex representations and their geodesic flows joint work with Martin Bridgeman, Dick Canary, Andres Sambarino François Labourie, Université Paris-Sud à Orsay, 16 September 2013 ICERM-Providence

2 Ingredients Gromov hyperbolic groups, boundary at infinity Gromov geodesic flow

3 Ingredients Gromov hyperbolic groups, boundary at infinity Gromov geodesic flow Convex Anosov representations contracted bundles over a flow limit curves Examples and properties the geodesic flow of a convex representation

4 Ingredients Gromov hyperbolic groups, boundary at infinity Gromov geodesic flow Convex Anosov representations contracted bundles over a flow limit curves Examples and properties the geodesic flow of a convex representation Metric anosov flows, (Smale flows) Stable (central) lamination Local product structure

5 Ingredients Gromov hyperbolic groups, boundary at infinity Gromov geodesic flow Convex Anosov representations contracted bundles over a flow limit curves Examples and properties the geodesic flow of a convex representation Metric anosov flows, (Smale flows) Stable (central) lamination Local product structure Main (embarassing) Theorem.

6 Ingredients Gromov hyperbolic groups, boundary at infinity Gromov geodesic flow Convex Anosov representations contracted bundles over a flow limit curves Examples and properties the geodesic flow of a convex representation Metric anosov flows, (Smale flows) Stable (central) lamination Local product structure Main (embarassing) Theorem. Elevating the ending lamination conjecture?

7 The Bowditch definition A non elementary hyperbolic group Γ has a boundary at infinity Γ which is a perfect metrizable compactum (= compact metric space without isolated points) on which Γ acts as a convergence group: the action on Γ 3 = {(x, y, z) Γ 3 x y z x}, is proper and cocompact.

8 The Bowditch definition A non elementary hyperbolic group Γ has a boundary at infinity Γ which is a perfect metrizable compactum (= compact metric space without isolated points) on which Γ acts as a convergence group: the action on Γ 3 = {(x, y, z) Γ 3 x y z x}, is proper and cocompact. Conversely, we can use these properties as definitions : Theorem [Bowditch] If Γ acts on a perfect metrizable compactum M as a convergence group then Γ is hyperbolic and M = Γ.

9 The Bowditch definition A non elementary hyperbolic group Γ has a boundary at infinity Γ which is a perfect metrizable compactum (= compact metric space without isolated points) on which Γ acts as a convergence group: the action on Γ 3 = {(x, y, z) Γ 3 x y z x}, is proper and cocompact. Conversely, we can use these properties as definitions : Theorem [Bowditch] If Γ acts on a perfect metrizable compactum M as a convergence group then Γ is hyperbolic and M = Γ. Example: Surface groups.

10 Gromov geodesic flow There exists a proper cocompact action of Γ on Ũ 0 Γ:= Γ 2 R commuting with the R action, unique up to reparametrisation once one imposes natural extra conditions. Gromov, Matheus, Champetier, Mineyev... then the corresponding action of R on U 0 Γ:= Ũ0Γ/Γ is called the Gromov geodesic flow

11 Gromov geodesic flow There exists a proper cocompact action of Γ on Ũ 0 Γ:= Γ 2 R commuting with the R action, unique up to reparametrisation once one imposes natural extra conditions. Gromov, Matheus, Champetier, Mineyev... then the corresponding action of R on U 0 Γ:= Ũ0Γ/Γ is called the Gromov geodesic flow Gromov, Coornaert Papadopoulos developed a symbolic coding for this flow which is finite to one.

12 Convex representation A representation ρ of a hyperbolic group Γ in SL(E) is convex if there exists continuous ρ-equivariant maps ξ and θ, called limit maps from Γ to P(E) and P(E ) respectively so that x y = ξ(x) ker(η(y)) = E.

13 Convex representation A representation ρ of a hyperbolic group Γ in SL(E) is convex if there exists continuous ρ-equivariant maps ξ and θ, called limit maps from Γ to P(E) and P(E ) respectively so that x y = ξ(x) ker(η(y)) = E. A construction: the associated flat bundle over U 0 Γ: ) E ρ := (Ũ0 E /Γ. decomposes, parallelly along the flow, as E ρ = Ξ Θ, with Ξ (x,y,t) := ξ(x) and Θ (x,y,t) := ker(θ(y)).

14 Convex Anosov representation Let M be a compact space quipped with a flow φ t and Φ t be a lift of φ t on some vector bundle F. Then F is contracted by the flow is T 0 > 0, u F, Φ T0 (u) 1 2 u.

15 Convex Anosov representation Let M be a compact space quipped with a flow φ t and Φ t be a lift of φ t on some vector bundle F. Then F is contracted by the flow is T 0 > 0, u F, Φ T0 (u) 1 2 u. A convex representation is Anosov, if Hom(Θ, Ξ) is contracted by the flow.

16 Convex Anosov representation Let M be a compact space quipped with a flow φ t and Φ t be a lift of φ t on some vector bundle F. Then F is contracted by the flow is T 0 > 0, u F, Φ T0 (u) 1 2 u. A convex representation is Anosov, if Hom(Θ, Ξ) is contracted by the flow. Convex Anosov Wanosov?

17 Examples Hitchin representations [Guichard Wienhard] (G, P)-Anosov representations: there exists a representation α of G so that if ρ is (G, P)-Anosov then α ρ is convex Anosov. [Guichard Wienhard] A convex irreducible representation is convex Anosov. Rank 1 convex cocompact convex Anosov. Benoist groups: acting cocompactly on a projective strict convex set. Small deformations of the above.

18 Properties Every matrix ρ(γ) is proximal: maximal eigenvalue λ ρ (γ) of multiplicity one / one attractive fixed point on P(E). Anosov=proximality spreads nicely The representation is well displacing A 1 d(γ) B λ ρ (γ) Ad(γ) + B, where d(γ):= inf η η.γ.η 1. [Delzant Guichard L Mozes] ρ is a quasiisometry. Injective, discrete image. [Kapovich Leeb Porti] have a more algebraic characterisation.

19 The geodesic flow of a convex representation Let Ũ ρ Γ:= {(u, v, x, y) E E Γ 2 u v = 1, u ξ(x), v θ(y)} We have an R-action given by (u, v, x, y) (t.u, t 1 v, x, y).

20 The geodesic flow of a convex representation Let Ũ ρ Γ:= {(u, v, x, y) E E Γ 2 u v = 1, u ξ(x), v θ(y)} We have an R-action given by (u, v, x, y) (t.u, t 1 v, x, y). Theorem [Geodesic flow for Convex Anosov] The action of Γ on ŨρΓ is proper and cocompact. The corresponding flow is orbit equivalent to the Gromov geodesic flow. Moreover the flow is a metric Anosov (Smale) flow.

21 Metric Anosov flow A lamination F= a foliation for a topological space. Two laminations define a product structure if in any small open sets they can be described as the two factors of a product.

22 Metric Anosov flow A lamination F= a foliation for a topological space. Two laminations define a product structure if in any small open sets they can be described as the two factors of a product. A flow φ t is metric Anosov if There exists two foliations F ± invariant by the flow, with product structure and F + and F are contracted towards the future and past respectively. y φ t(x) u φ t(z) φt(y) x φt(u) z

23 Embarassements The result is useful and a necessary step to obtain to obtain a 1-1 coding and use the thermodynamical formalism. But:

24 Embarassements The result is useful and a necessary step to obtain to obtain a 1-1 coding and use the thermodynamical formalism. But: We do not know of hyperbolic groups admitting convex representations which are not abstractly rank 1- convex cocompact groups or Benoist groups (in which case the Anosov character of the geodesic flow is well known)

25 Embarassements The result is useful and a necessary step to obtain to obtain a 1-1 coding and use the thermodynamical formalism. But: We do not know of hyperbolic groups admitting convex representations which are not abstractly rank 1- convex cocompact groups or Benoist groups (in which case the Anosov character of the geodesic flow is well known) Does there exists a hyperbolic group whose geodesic flow is not Anosov?

26 Elevating the ending lamination conjecture? What can we say of representations ρ which are limits of convex Anosov representations? In particular for the Barbot component of SL(3, R)? Definition: without incidental parabolics:= being limits+all ρ(γ) are proximal.

27 Elevating the ending lamination conjecture? What can we say of representations ρ which are limits of convex Anosov representations? In particular for the Barbot component of SL(3, R)? Definition: without incidental parabolics:= being limits+all ρ(γ) are proximal. Question: Without incidental parabolics + quasiisometry = convex Anosov for surface groups?

28 Elevating the ending lamination conjecture? What can we say of representations ρ which are limits of convex Anosov representations? In particular for the Barbot component of SL(3, R)? Definition: without incidental parabolics:= being limits+all ρ(γ) are proximal. Question: Without incidental parabolics + quasiisometry = convex Anosov for surface groups? Question: Do representations without incidental parabolics but not convex Anosov exists? Yes (?) for SL(n, C) and complex groups? But what about SL(3, R)?

29 Elevating the ending lamination conjecture? What can we say of representations ρ which are limits of convex Anosov representations? In particular for the Barbot component of SL(3, R)? Definition: without incidental parabolics:= being limits+all ρ(γ) are proximal. Question: Without incidental parabolics + quasiisometry = convex Anosov for surface groups? Question: Do representations without incidental parabolics but not convex Anosov exists? Yes (?) for SL(n, C) and complex groups? But what about SL(3, R)? Question: Existence of Cannon Thurston maps?

30 Elevating the ending lamination conjecture? What can we say of representations ρ which are limits of convex Anosov representations? In particular for the Barbot component of SL(3, R)? Definition: without incidental parabolics:= being limits+all ρ(γ) are proximal. Question: Without incidental parabolics + quasiisometry = convex Anosov for surface groups? Question: Do representations without incidental parabolics but not convex Anosov exists? Yes (?) for SL(n, C) and complex groups? But what about SL(3, R)? Question: Existence of Cannon Thurston maps? Question: How to associate ending invariants to such representations?

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