Superrigidity and Measure Equivalence, Part I Alex Furman - PowerPoint PPT Presentation

Superrigidity and Measure Equivalence, Part I Alex Furman University of Illinois at Chicago Institut Henri Poincar´ e, Paris, June 20 2011 1/14

Poincar´ e disc and surfaces 2/14

Poincar´ e disc and surfaces The simplest simple Lie group G ◮ SL 2 ( R ) ◮ PSL 2 ( R ) = Isom + ( H 2 ) ◮ PGL 2 ( R ) = Isom( H 2 ) 2/14

Poincar´ e disc and surfaces The simplest simple Lie group G ◮ SL 2 ( R ) ◮ PSL 2 ( R ) = Isom + ( H 2 ) ◮ PGL 2 ( R ) = Isom( H 2 ) with H 2 = G / K where K ≃ SO 2 ( R ) 2/14

Poincar´ e disc and surfaces The simplest simple Lie group G ◮ SL 2 ( R ) ◮ PSL 2 ( R ) = Isom + ( H 2 ) ◮ PGL 2 ( R ) = Isom( H 2 ) with H 2 = G / K where K ≃ SO 2 ( R ) Fix a closed surface Σ be of genus ≥ 2 (up to Diff(Σ) 0 ) By uniformization, ∃ (many) Riemannian g on Σ with K ≡ − 1 . 2/14

Poincar´ e disc and surfaces The simplest simple Lie group G ◮ SL 2 ( R ) ◮ PSL 2 ( R ) = Isom + ( H 2 ) ◮ PGL 2 ( R ) = Isom( H 2 ) with H 2 = G / K where K ≃ SO 2 ( R ) Fix a closed surface Σ be of genus ≥ 2 (up to Diff(Σ) 0 ) By uniformization, ∃ (many) Riemannian g on Σ with K ≡ − 1 . � a Riemannian covering p : H 2 → (Σ , g ), unique up to G = Isom( H 2 ) 2/14

Poincar´ e disc and surfaces The simplest simple Lie group G ◮ SL 2 ( R ) ◮ PSL 2 ( R ) = Isom + ( H 2 ) ◮ PGL 2 ( R ) = Isom( H 2 ) with H 2 = G / K where K ≃ SO 2 ( R ) Fix a closed surface Σ be of genus ≥ 2 (up to Diff(Σ) 0 ) By uniformization, ∃ (many) Riemannian g on Σ with K ≡ − 1 . � a Riemannian covering p : H 2 → (Σ , g ), unique up to G = Isom( H 2 ) � an embedding Γ = π 1 (Σ , ∗ ) → G , unique up to G -conjugation 2/14

Poincar´ e disc and surfaces The simplest simple Lie group G ◮ SL 2 ( R ) ◮ PSL 2 ( R ) = Isom + ( H 2 ) ◮ PGL 2 ( R ) = Isom( H 2 ) with H 2 = G / K where K ≃ SO 2 ( R ) Fix a closed surface Σ be of genus ≥ 2 (up to Diff(Σ) 0 ) By uniformization, ∃ (many) Riemannian g on Σ with K ≡ − 1 . � a Riemannian covering p : H 2 → (Σ , g ), unique up to G = Isom( H 2 ) � an embedding Γ = π 1 (Σ , ∗ ) → G , unique up to G -conjugation Defn: Teichm¨ uller space = moduli of hyperbolic metrics on Σ Teich(Σ) = { lattice embeddings ρ : Γ → G } / G 2/14

Flexibility of lattices in SL 2 ( R ) Theorem (Riemann ?, Poincar´ e, Teichm¨ uller ?) For a closed surface of genus g ≥ 2 one has Teich(Σ) ∼ = R 6 · g − 6 3/14

Flexibility of lattices in SL 2 ( R ) Theorem (Riemann ?, Poincar´ e, Teichm¨ uller ?) For a closed surface of genus g ≥ 2 one has Teich(Σ) ∼ = R 6 · g − 6 There are R 6 g − 6 many G -conjugacy classes of lattice embeddings Γ → G = PSL 2 ( R ) where Γ = � a 1 , . . . , a g , b 1 , . . . , b g | [ a 1 , b 1 ] · · · [ a g , b g ] = 1 � 3/14

Flexibility of lattices in SL 2 ( R ) Theorem (Riemann ?, Poincar´ e, Teichm¨ uller ?) For a closed surface of genus g ≥ 2 one has Teich(Σ) ∼ = R 6 · g − 6 There are R 6 g − 6 many G -conjugacy classes of lattice embeddings Γ → G = PSL 2 ( R ) where Γ = � a 1 , . . . , a g , b 1 , . . . , b g | [ a 1 , b 1 ] · · · [ a g , b g ] = 1 � Remarks ◮ ∀ ρ 1 , ρ 2 : Γ → PSL 2 ( R ) lattice embeddings ∃ ! f ∈ Homeo( S 1 = ∂ H 2 ) ρ 2 ( γ ) = f − 1 ◦ ρ 1 ( γ ) ◦ f ◮ Similar results apply to non-uniform lattices. 3/14

Mostow’s strong rigidity 4/14

Mostow’s strong rigidity Theorem 1 (Mostow ’68) ◮ A closed manifold M n of dim n ≥ 3 admits at most one hyperbolic metric. 4/14

� � � Mostow’s strong rigidity Theorem 1 (Mostow ’68) ◮ A closed manifold M n of dim n ≥ 3 admits at most one hyperbolic metric. ◮ G = Isom( H n ), n ≥ 3, and Γ , Γ ′ < G uniform lattices = Γ ′ there Given j : Γ ∼ j ( γ ) = g − 1 γ g . ∃ ! g ∈ G with ∼ = Γ Γ ′ ⊂ ⊂ ∼ = � G ′ G 4/14

� � � Mostow’s strong rigidity Theorem 1 (Mostow ’68) ◮ A closed manifold M n of dim n ≥ 3 admits at most one hyperbolic metric. ◮ G = Isom( H n ), n ≥ 3, and Γ , Γ ′ < G uniform lattices = Γ ′ there Given j : Γ ∼ j ( γ ) = g − 1 γ g . ∃ ! g ∈ G with Theorem 2 (Mostow) G = Isom( H ), G ′ = Isom( H ′ ) where H , H ′ ∈ { H n , H n C , H n H , H 2 O } \ H 2 . Let Γ < G , Γ ′ < G ′ be uniform lattices and j : Γ ∼ = Γ ′ an isomorphism. = Γ ′ extends to an isomorphism G ∼ Then j : Γ ∼ = G ′ . ∼ = Γ Γ ′ ⊂ ⊂ ∼ = � G ′ G 4/14

� � � Mostow’s strong rigidity Theorem 1 (Mostow ’68) ◮ A closed manifold M n of dim n ≥ 3 admits at most one hyperbolic metric. ◮ G = Isom( H n ), n ≥ 3, and Γ , Γ ′ < G uniform lattices = Γ ′ there Given j : Γ ∼ j ( γ ) = g − 1 γ g . ∃ ! g ∈ G with Theorem 2 (Mostow) G = Isom( H ), G ′ = Isom( H ′ ) where H , H ′ ∈ { H n , H n C , H n H , H 2 O } \ H 2 . Let Γ < G , Γ ′ < G ′ be uniform lattices and j : Γ ∼ = Γ ′ an isomorphism. = Γ ′ extends to an isomorphism G ∼ Then j : Γ ∼ = G ′ . ∼ = Theorem 3 (Mostow ’73) Γ Γ ′ Same for any (semi)-simple G , G ′ �≃ SL 2 ( R ) ⊂ ⊂ and uniform (irreducible) lattices Γ < G , Γ ′ < G ′ . ∼ = � G ′ G 4/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 Show that f is quasi-conformal and improve to conformal (using n ≥ 3). 2 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 Show that f is quasi-conformal and improve to conformal (using n ≥ 3). 2 Quasi-isometry: a map q : X → Y s.t. ∃ K , A , C K − 1 · d X ( x , x ′ ) − A < d Y ( q ( x ) , q ( x ′ )) < K · d X ( x , x ′ ) + A ◮ 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 Show that f is quasi-conformal and improve to conformal (using n ≥ 3). 2 Quasi-isometry: a map q : X → Y s.t. ∃ K , A , C K − 1 · d X ( x , x ′ ) − A < d Y ( q ( x ) , q ( x ′ )) < K · d X ( x , x ′ ) + A ◮ ∀ y ∈ Y , ∃ x ∈ X , d ( q ( x ) , y ) < C . ◮ 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 Show that f is quasi-conformal and improve to conformal (using n ≥ 3). 2 Quasi-isometry: a map q : X → Y s.t. ∃ K , A , C K − 1 · d X ( x , x ′ ) − A < d Y ( q ( x ) , q ( x ′ )) < K · d X ( x , x ′ ) + A ◮ ∀ y ∈ Y , ∃ x ∈ X , d ( q ( x ) , y ) < C . ◮ Step 1 of Mostow’s proof ◮ ∃ quasi-isometry q : H n → Cayley (Γ , S ) = Cayley (Γ ′ , j ( S )) → H n 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 Show that f is quasi-conformal and improve to conformal (using n ≥ 3). 2 Quasi-isometry: a map q : X → Y s.t. ∃ K , A , C K − 1 · d X ( x , x ′ ) − A < d Y ( q ( x ) , q ( x ′ )) < K · d X ( x , x ′ ) + A ◮ ∀ y ∈ Y , ∃ x ∈ X , d ( q ( x ) , y ) < C . ◮ Step 1 of Mostow’s proof ◮ ∃ quasi-isometry q : H n → Cayley (Γ , S ) = Cayley (Γ ′ , j ( S )) → H n ◮ Any quasi-isometry q : H n → H n extends to a qc-homeo f : ∂ H n → ∂ H n 5/14

Sketch of Mostow’s proof of Theorem 1 Given: ◮ Γ , Γ ′ � H n properly discontinuous cocompact isometric actions. ◮ An isomorphism of abstract groups j : Γ ∼ = Γ ′ Show: ∃ a homeomorphism f : ∂ H n → ∂ H n so that f ( γξ ) = j ( γ ) f ( ξ ). 1 Show that f is quasi-conformal and improve to conformal (using n ≥ 3). 2 Quasi-isometry: a map q : X → Y s.t. ∃ K , A , C K − 1 · d X ( x , x ′ ) − A < d Y ( q ( x ) , q ( x ′ )) < K · d X ( x , x ′ ) + A ◮ ∀ y ∈ Y , ∃ x ∈ X , d ( q ( x ) , y ) < C . ◮ Step 1 of Mostow’s proof ◮ ∃ quasi-isometry q : H n → Cayley (Γ , S ) = Cayley (Γ ′ , j ( S )) → H n ◮ Any quasi-isometry q : H n → H n extends to a qc-homeo f : ∂ H n → ∂ H n ◮ f is j -equivariant 5/14