Complete flat Lorentzian three-manifolds Introduction Proper group - PowerPoint PPT Presentation

Complete flat Lorentzian three-manifolds J. Danciger Complete flat Lorentzian three-manifolds Introduction Proper group actions AdS geometry: fibrations A new properness criterion Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, geometric transition Jeffrey Danciger with F . Gu´ eritaud and F . Kassel University of Texas – Austin jdanciger@math.utexas.edu XXII nd Rolf Nevanlinna Colloquium August 8, 2013

Complete flat Lorentzian manifolds Complete flat Lorentzian three-manifolds J. Danciger Question Introduction 1. Classify the topology of manifolds M admitting a flat Lorentzian metric, Proper group actions which is geodesically complete. AdS geometry: fibrations 2. Understand the geometry of those metrics. A new properness criterion These manifolds are quotients of Minkowski space R n , 1 : M = Γ \ R n , 1 , where Fibrations of Margulis Γ ⊂ Isom ( R n , 1 ) = O ( n , 1 ) ⋉ R n + 1 , a group of isometries. spacetimes Extra Slides: proof of Question properness criterion, geometric transition 3. Which groups Γ can act properly discontinuosly and freely by isometries on Minowski space? 4. Given such Γ , classify proper isometric group actions Γ → Isom ( R n , 1 ) . Study dimension n = 3. 3. long known due to Fried–Goldman 4. nearly complete (Fried–Goldman, Goldman–Labourie–Margulis, D–Gu´ eritaud–Kassel) Today: give a topological classification ( Γ finitely generated); in particular all complete flat Lorentian three-manifolds are tame .

Which groups act properly discontinuously? Complete flat Lorentzian three-manifolds Let Γ ⊂ O ( 2 , 1 ) ⋉ R 3 be a group of isometries. J. Danciger Then, T ∈ Γ is of the form T ( x ) = Ax + b Introduction A ∈ O ( 2 , 1 ) is the linear part and b ∈ R 3 is the translational part . Proper group actions Denote the homomorphism L : T �→ A . AdS geometry: fibrations Easy example: Γ = Z × Z × Z , generated by three independent translations. A new properness criterion Here, the linear part L (Γ) is trivial and M = Γ \ R 3 ∼ = T 3 . Fibrations of Margulis spacetimes Fried–Goldman ’83: If Γ acts properly and freely, then either Extra Slides: proof of properness criterion, ◮ Γ is solvable, or geometric transition ◮ L : Γ → O ( 2 , 1 ) is injective and discrete. So Γ ∼ = L (Γ) is a surface group, necessarily non-compact (Mess). = ⇒ Γ is a free group. In fact, dichotomy holds more generally for Γ ⊂ Aff ( R 3 ) . Conjecture (Auslander ’64) A group of affine transformations acting properly discontinuously and cocompactly on affine space is virtually solvable.

Free groups acting properly discontinuously Complete flat Lorentzian three-manifolds Fried–Goldman ’83: If Γ acts properly, freely and Γ is not solvable then the J. Danciger linear part L (Γ) ∼ = Γ is a non-compact surface group. = ⇒ Γ a free group. Introduction Proper group actions AdS geometry: ◮ Γ is called affine deformation of L (Γ) . fibrations ◮ Margulis ’83: first examples. A new properness criterion ◮ M called a Margulis spacetime . Fibrations of Margulis spacetimes Extra Slides: proof of properness criterion, So, in n = 3, it remains to: geometric transition ◮ Understand which affine deformations of a surface group give proper actions. ◮ Determine the topology of the quotients.

Topology: tameness Complete flat Lorentzian three-manifolds Let the free group Γ ⊂ O ( 2 , 1 ) ⋉ R 2 , 1 act properly discontinuously on R 2 , 1 . J. Danciger Let M = Γ \ R 2 , 1 be the quotient manifold. Introduction Proper group actions ◮ Expect M to be a handle-body. But is M even tame? AdS geometry: ◮ A manifold is tame if it is the interior of a compact manifold with boundary. fibrations ◮ Exist wild manifolds which are homotopically trivial (Whitehead). A new properness criterion ◮ Tameness of hyperbolic three-manifolds: conjectured by Marden in ’74. only Fibrations of Margulis recently proven by Agol and Calegari–Gabai in ’06. spacetimes ◮ Drumm (’90) constructed examples Γ by building fundamental domains Extra Slides: proof of (crooked planes) ( = ⇒ tame). properness criterion, geometric transition ◮ Drumm–Goldman conjectured that all Margulis spacetimes can be built by Drumm’s technique. therefore Conjecture (Drumm-Goldman ’90s) All Margulis spacetimes are tame ◮ Partial results due to Charette–Drumm–Goldman, in case rank (Γ) = 2.

Topology: tameness Complete flat Lorentzian three-manifolds J. Danciger Theorem (D–Gu´ eritaud–Kassel, Choi–Goldman) Let Γ ⊂ O ( 2 , 1 ) ⋉ R 3 be a free group which acts properly on R 2 , 1 . Assume the Introduction linear part L (Γ) is convex cocompact. Then the quotient M = Γ \ R 2 , 1 is tame, Proper group actions diffeomorphic to a handle-body. AdS geometry: fibrations A new properness ◮ In progress: general case allowing for linear part L (Γ) to have parabolics. criterion will classify the topology of all complete affine three-manifolds. Fibrations of Margulis spacetimes ◮ We prove that M is fibered in (time-like) geodesics over a hyperbolic Extra Slides: proof of surface S = L (Γ) \ H 2 , where L (Γ) is the linear part. properness criterion, ⇒ M ∼ geometric transition = = S × R . ◮ point of view: Margulis spacetimes (flat) behave like infinitesimal versions of AdS manifolds (curved). ◮ AdS = anti de Sitter geometry: constant negative curvature model space. ◮ can make the intuition precise: (paraphrased) Theorem (D-Gu´ eritaud-Kassel) The complete flat Lorentzian metric on M is realized as a limit of complete metrics of constant curvature.

Hyperbolic surfaces Complete flat Lorentzian three-manifolds H 2 = upper hemisphere in CP 1 . J. Danciger Isom 0 H 2 = PSL ( 2 , R ) =: G . Introduction Proper group actions ◮ Γ 0 ⊂ PSL ( 2 , R ) discrete = ⇒ Γ 0 acts properly discontinuously. AdS geometry: ◮ b/c point stabilizer is compact. fibrations ◮ If Γ 0 is f.g. and acts freely then S = Γ 0 \ H 2 is a surface of finite type A new properness criterion (tameness in two-dimensions). Fibrations of Margulis ◮ Γ 0 called convex cocompact if all elements are hyperbolic (translation spacetimes length > 0). Extra Slides: proof of properness criterion, geometric transition

Deforming representations Complete flat Lorentzian three-manifolds G = PSL ( 2 , R ) . J. Danciger g = sl ( 2 , R ) . Introduction Proper group actions Consider a path ρ t : Γ 0 → G with ρ 0 = id : Γ 0 → Γ 0 . AdS geometry: fibrations � ◮ Derivative ρ ′ = d is a representation Γ 0 → TG ∼ � = G ⋉ g . � A new properness d t � t = 0 criterion Fibrations of Margulis = ⇒ ρ ′ ( γ ) = ( γ, u ( γ )) where u : Γ 0 → g is spacetimes � Extra Slides: proof of u ( γ ) = d � ρ t ( γ ) γ − 1 properness criterion, � d t geometric transition � t = 0 “infinitesimal deformation” of Γ 0 in G . cocycle condition: u ( γ 1 γ 2 ) = u ( γ 1 ) + Ad γ 1 u ( γ 2 ) . ◮ γ �→ ( γ, u ( γ )) is a representation Γ 0 → G ⋉ g . Notation Γ u := { ( γ, u ( γ )) } ⊂ G ⋉ g . 0

The associated affine action Complete flat Lorentzian three-manifolds J. Danciger Notation Introduction Γ u := { ( γ, u ( γ )) } ⊂ G ⋉ g . Proper group actions 0 AdS geometry: fibrations A new properness G ⋉ g acts by affine transformations on g : criterion Fibrations of Margulis ( g , u ) ∈ G ⋉ g acts by ( g , u ) · v := Ad g v + u spacetimes Extra Slides: proof of properness criterion, geometric transition ◮ preserves indefinite metric on g coming from Killing form . ◮ For G = PSL ( 2 , R ) , signature = ( 2 , 1 ) ⇒ identify R 2 , 1 = g . = ◮ ◮ therefore G ⋉ g ⊂ Isom R 2 , 1 . ◮ In fact, G ⋉ g = Isom 0 R 2 , 1 . ◮ by Fried–Goldman, any proper isometric action on R 2 , 1 must have the form Γ u 0 (up to f.i.). Which Γ u 0 act properly?

Margulis’s opposite sign lemma Complete flat Lorentzian three-manifolds exponential decay J. Danciger exponential growth Introduction Proper group actions Action of ( γ, u ( γ )) : linear translation AdS geometry: fibrations A new properness criterion Fibrations of Margulis Margulis invariant: spacetimes Extra Slides: proof of Observation properness criterion, geometric transition signed translation length of ( γ, u ( γ )) given by d λ u ( γ ) . Notation ◮ λ ( γ ) = hyperbolic translation length of γ ∈ PSL ( 2 , R ) . ◮ d λ u ( γ ) = rate of change of λ ( γ ) under the infinitesimal deformation u ( γ ) . Lemma (Margulis) If there are γ 1 , γ 2 ∈ Γ 0 such that d λ u ( γ 1 ) ≥ 0 and d λ u ( γ 2 ) ≤ 0 then Γ u 0 does not act properly discontinuously on R 2 , 1 .