Lattice Envelopes Uri Bader Alex Furman Roman Sauer The Technion, - PowerPoint PPT Presentation

Lattice Envelopes Uri Bader Alex Furman Roman Sauer The Technion, Haifa University of Illinois at Chicago Universit¨ at Regensburg AMS Special Meeting, 2010-11-05 1/11

Lattice Envelopes Definition ◮ A subgroup Γ < G is a lattice in a lcsc group G if Γ is discrete and Haar ( G / Γ) < ∞ . 2/11

Lattice Envelopes Definition ◮ A subgroup Γ < G is a lattice in a lcsc group G if Γ is discrete and Haar ( G / Γ) < ∞ . Equivalently, Γ has a Borel fundamental domain F ⊂ G with m G ( F ) < ∞ . 2/11

Lattice Envelopes Definition ◮ A subgroup Γ < G is a lattice in a lcsc group G if Γ is discrete and Haar ( G / Γ) < ∞ . Equivalently, Γ has a Borel fundamental domain F ⊂ G with m G ( F ) < ∞ . ◮ A lattice Γ < G is uniform if G / Γ is compact, non-uniform otherwise. 2/11

Lattice Envelopes Definition ◮ A subgroup Γ < G is a lattice in a lcsc group G if Γ is discrete and Haar ( G / Γ) < ∞ . Equivalently, Γ has a Borel fundamental domain F ⊂ G with m G ( F ) < ∞ . ◮ A lattice Γ < G is uniform if G / Γ is compact, non-uniform otherwise. i ◮ A homomorphism Γ − → G is a lattice embedding if i (Γ) < G is a lattice and | Ker ( i ) | < ∞ . 2/11

Lattice Envelopes Definition ◮ A subgroup Γ < G is a lattice in a lcsc group G if Γ is discrete and Haar ( G / Γ) < ∞ . Equivalently, Γ has a Borel fundamental domain F ⊂ G with m G ( F ) < ∞ . ◮ A lattice Γ < G is uniform if G / Γ is compact, non-uniform otherwise. i ◮ A homomorphism Γ − → G is a lattice embedding if i (Γ) < G is a lattice and | Ker ( i ) | < ∞ . Problem i Given Γ , describe all lattice envelopes: groups G with a lattice embedding Γ − → G. 2/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) Id Trivial lattice Γ − → Γ 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) Id Trivial lattice Γ − → Γ Constructions i Let Γ − → G be a lattice embedding. 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) Id Trivial lattice Γ − → Γ Constructions i Let Γ − → G be a lattice embedding. Then → G ′ is a lattice imbedding for Γ ′ = Γ ∩ i − 1 ( G ′ ) i ◮ If [ G : G ′ ] < ∞ , then Γ ′ − 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) Id Trivial lattice Γ − → Γ Constructions i Let Γ − → G be a lattice embedding. Then → G ′ is a lattice imbedding for Γ ′ = Γ ∩ i − 1 ( G ′ ) i ◮ If [ G : G ′ ] < ∞ , then Γ ′ − i ◮ If K ⊳ G is compact, then Γ − → G − → G / K is a lattice imbedding 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) Id Trivial lattice Γ − → Γ Constructions i Let Γ − → G be a lattice embedding. Then → G ′ is a lattice imbedding for Γ ′ = Γ ∩ i − 1 ( G ′ ) i ◮ If [ G : G ′ ] < ∞ , then Γ ′ − i ◮ If K ⊳ G is compact, then Γ − → G − → G / K is a lattice imbedding i ◮ If i (Γ) < H < G a closed subgroup, then Γ − → H is a lattice imbedding 3/11

Basic Examples Examples Classical lattices in s-s real Lie groups: π 1 (Σ g ) < PSL 2 ( R ), PSL n ( Z ) < PSL n ( R ) S -arithmetic SL n ( Z [ 1 p ]) < SL n ( R ) × SL n ( Q p ) Combinatorial π 1 ( X ) < Aut(˜ X ), X fin simpl cpx, e.g., F n < Aut( T 2 n ) Id Trivial lattice Γ − → Γ Constructions i Let Γ − → G be a lattice embedding. Then → G ′ is a lattice imbedding for Γ ′ = Γ ∩ i − 1 ( G ′ ) i ◮ If [ G : G ′ ] < ∞ , then Γ ′ − i ◮ If K ⊳ G is compact, then Γ − → G − → G / K is a lattice imbedding i ◮ If i (Γ) < H < G a closed subgroup, then Γ − → H is a lattice imbedding ◮ If Λ < H is a lattice imbedding, then Γ × Λ < G × H is a lattice imbedding. 3/11

The case of Free groups Example Some lattice embeddings of Γ = F n , 1 < n < ∞ : 4/11

The case of Free groups Example Some lattice embeddings of Γ = F n , 1 < n < ∞ : Γ < PSL 2 ( R ) (non-uniform) 1 Γ < PSL 2 ( Q p ) (uniform) 2 Γ < Aut( T 2 n ) (uniform). 3 4/11

The case of Free groups Example Some lattice embeddings of Γ = F n , 1 < n < ∞ : Γ < PSL 2 ( R ) (non-uniform) 1 Γ < PSL 2 ( Q p ) (uniform) 2 Γ < Aut( T 2 n ) (uniform). 3 Theorem Let F n − → G be a lattice embedding (uniform or non-uniform). Then ◮ either K − → G − → PSL 2 ( R ) or PGL 2 ( R ) ◮ or K − → G − → H where H < Aut( T ) cocompact action on a bdd deg tree according to whether F n < G is non-uniform or uniform lattice imbedding. 4/11

The case of Free groups Example Some lattice embeddings of Γ = F n , 1 < n < ∞ : Γ < PSL 2 ( R ) (non-uniform) 1 Γ < PSL 2 ( Q p ) (uniform) 2 Γ < Aut( T 2 n ) (uniform). 3 Theorem Let F n − → G be a lattice embedding (uniform or non-uniform). Then ◮ either K − → G − → PSL 2 ( R ) or PGL 2 ( R ) ◮ or K − → G − → H where H < Aut( T ) cocompact action on a bdd deg tree according to whether F n < G is non-uniform or uniform lattice imbedding. The uniform case uses a result of Mosher - Sageev - Whyte. 4/11

The case of classical lattices Theorem (Rigidity of Classical lattices, extends [F. 2001] ) Let F n �≃ Γ < H be (irred.) lattice in a conn, center free, (semi)-simple real Lie group H w/o compact factors. Let Γ − → G be a lattice imbedding. 5/11

The case of classical lattices Theorem (Rigidity of Classical lattices, extends [F. 2001] ) Let F n �≃ Γ < H be (irred.) lattice in a conn, center free, (semi)-simple real Lie group H w/o compact factors. Let Γ − → G be a lattice imbedding. Then ◮ either, up to fin ind and compact kernel G is H, or ◮ or up to fin ind and compact kernel G is Γ . 5/11

The case of classical lattices Theorem (Rigidity of Classical lattices, extends [F. 2001] ) Let F n �≃ Γ < H be (irred.) lattice in a conn, center free, (semi)-simple real Lie group H w/o compact factors. Let Γ − → G be a lattice imbedding. Then ◮ either, up to fin ind and compact kernel G is H, or ◮ or up to fin ind and compact kernel G is Γ . Example Γ = PSL 2 ( Z [ 1 p ]) < G = PSL 2 ( R ) × H where PSL 2 ( Q p ) < H < Aut( T p +1 ). 5/11

The case of classical lattices Theorem (Rigidity of Classical lattices, extends [F. 2001] ) Let F n �≃ Γ < H be (irred.) lattice in a conn, center free, (semi)-simple real Lie group H w/o compact factors. Let Γ − → G be a lattice imbedding. Then ◮ either, up to fin ind and compact kernel G is H, or ◮ or up to fin ind and compact kernel G is Γ . Example Γ = PSL 2 ( Z [ 1 p ]) < G = PSL 2 ( R ) × H where PSL 2 ( Q p ) < H < Aut( T p +1 ). Theorem (Rigidity of S -arithmetic lattices) Let Γ < H = H ( ∞ ) × H ( fin ) be an S-arithmetic lattice H ( k ( S )) < � ν ∈ S H ( k ν ) . Let Γ → G be a lattice imbedding. Then up to fin ind and compact kernel ◮ either G is H ( ∞ ) × H ( fin ) , ∗ , where H ( fin ) < H ( fin ) , ∗ < Aut( X B − T ) ◮ or G is Γ . 5/11

The case of convergence groups Convergence groups Group Γ is a convergence group if there is a minimal action Γ → Homeo( M ) with infinite compact M so that the action on M 3 \ ∆ is proper. 6/11