The space of metrics on Gromov hyperbolic groups Alex Furman - PowerPoint PPT Presentation

The space of metrics on Gromov hyperbolic groups Alex Furman University of Illinois at Chicago Northwestern University, 2010-10-31 1/13

Negatively Curved Manifolds: Geometry Setting ( M , g ) where M - closed manifold, g - Riemannian metric with K < 0 2/13

Negatively Curved Manifolds: Geometry Setting ( M , g ) where M - closed manifold, g - Riemannian metric with K < 0 Marked Length Spectrum ◮ Free homotopy classes [ S 1 ; M ] = { S 1 → M } / ∼ . 2/13

Negatively Curved Manifolds: Geometry Setting ( M , g ) where M - closed manifold, g - Riemannian metric with K < 0 Marked Length Spectrum ◮ Free homotopy classes [ S 1 ; M ] = { S 1 → M } / ∼ . ◮ ∀ c 0 � = c ∈ [ S 1 ; M ] , ∃ ! closed geodesic geo c in c . 2/13

Negatively Curved Manifolds: Geometry Setting ( M , g ) where M - closed manifold, g - Riemannian metric with K < 0 Marked Length Spectrum ◮ Free homotopy classes [ S 1 ; M ] = { S 1 → M } / ∼ . ◮ ∀ c 0 � = c ∈ [ S 1 ; M ] , ∃ ! closed geodesic geo c in c . ◮ Marked Length Spectrum: c �→ ℓ g ( c ) = Length g (geo c ). 2/13

Negatively Curved Manifolds: Geometry Setting ( M , g ) where M - closed manifold, g - Riemannian metric with K < 0 Marked Length Spectrum ◮ Free homotopy classes [ S 1 ; M ] = { S 1 → M } / ∼ . ◮ ∀ c 0 � = c ∈ [ S 1 ; M ] , ∃ ! closed geodesic geo c in c . ◮ Marked Length Spectrum: c �→ ℓ g ( c ) = Length g (geo c ). Marked Length Spectrum Rigidity ◮ Conjecture (Burns-Katok ’85): ℓ g determines g , up to Diff( M ) 0 2/13

Negatively Curved Manifolds: Geometry Setting ( M , g ) where M - closed manifold, g - Riemannian metric with K < 0 Marked Length Spectrum ◮ Free homotopy classes [ S 1 ; M ] = { S 1 → M } / ∼ . ◮ ∀ c 0 � = c ∈ [ S 1 ; M ] , ∃ ! closed geodesic geo c in c . ◮ Marked Length Spectrum: c �→ ℓ g ( c ) = Length g (geo c ). Marked Length Spectrum Rigidity ◮ Conjecture (Burns-Katok ’85): ℓ g determines g , up to Diff( M ) 0 ◮ Deformation rigidity (Guillemin-Kazhdan ’80) ◮ Surfaces (Otal ’90, Croke ’90) ◮ ( M , g ) loc. symmetric (Hamenst¨ adt ’99, using BCG) 2/13

Negatively Curved manifolds: Dynamics of ( SM , φ t ) ◮ Topological entropy h top of φ t on SM 3/13

Negatively Curved manifolds: Dynamics of ( SM , φ t ) ◮ Topological entropy h top of φ t on SM ◮ Stable/Unstable foliations 3/13

Negatively Curved manifolds: Dynamics of ( SM , φ t ) ◮ Topological entropy h top of φ t on SM ◮ Stable/Unstable foliations ◮ Bowen-Margulis measure µ BM on SM 3/13

Negatively Curved manifolds: Dynamics of ( SM , φ t ) ◮ Topological entropy h top of φ t on SM ◮ Stable/Unstable foliations ◮ Bowen-Margulis measure µ BM on SM which is the unique measure of maximal entropy: 1 Ent ( SM , φ t , µ BM ) = h top 3/13

Negatively Curved manifolds: Dynamics of ( SM , φ t ) ◮ Topological entropy h top of φ t on SM ◮ Stable/Unstable foliations ◮ Bowen-Margulis measure µ BM on SM which is the unique measure of maximal entropy: 1 Ent ( SM , φ t , µ BM ) = h top is weal limit of periodic orbits organized by length 2 1 X µ BM = lim # { c | ℓ ( c ) < T } · λ (geo c ) T →∞ { c | ℓ ( c ) < T } 3/13

Negatively Curved manifolds: Dynamics of ( SM , φ t ) ◮ Topological entropy h top of φ t on SM ◮ Stable/Unstable foliations ◮ Bowen-Margulis measure µ BM on SM which is the unique measure of maximal entropy: 1 Ent ( SM , φ t , µ BM ) = h top is weal limit of periodic orbits organized by length 2 1 X µ BM = lim # { c | ℓ ( c ) < T } · λ (geo c ) T →∞ { c | ℓ ( c ) < T } has conditionals on stable/unstable scaled by e ± ht where h = h top 3 BM = e − ht · d µ ( s ) BM = e + ht · d µ ( u ) d φ t ∗ µ ( s ) d φ t ∗ µ ( u ) BM , BM 3/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) ◮ F.h.c. [ S 1 ; M ] are conj classes: 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e g on ˜ ◮ g on M Γ-invariant metric ˜ M � 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e g on ˜ ◮ g on M Γ-invariant metric ˜ M � ℓ g ( � γ � ) = min dist ˜ g ( γ · x , x ) x ∈ ˜ M 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e g on ˜ ◮ g on M Γ-invariant metric ˜ M � 1 g ( γ n y , y ) ℓ g ( � γ � ) = min dist ˜ g ( γ · x , x ) = lim n dist ˜ x ∈ ˜ n →∞ M 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e g on ˜ ◮ g on M Γ-invariant metric ˜ M � 1 g ( γ n y , y ) ℓ g ( � γ � ) = min dist ˜ g ( γ · x , x ) = lim n dist ˜ x ∈ ˜ n →∞ M ◮ Top entropy = volume entropy = Γ-orbit growth 1 h top = lim R log vol ˜ g ( B x , R ) R →∞ 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e g on ˜ ◮ g on M Γ-invariant metric ˜ M � 1 g ( γ n y , y ) ℓ g ( � γ � ) = min dist ˜ g ( γ · x , x ) = lim n dist ˜ x ∈ ˜ n →∞ M ◮ Top entropy = volume entropy = Γ-orbit growth 1 1 h top = lim R log vol ˜ g ( B x , R ) = lim R log #(Γ · y ∩ B x , R ) R →∞ R →∞ 4/13

Negatively Curved manifolds Inside-Out ◮ Instead of M think of � M or better Γ = π 1 ( M , x ) � � ◮ F.h.c. [ S 1 ; M ] are conj classes: C Γ = � γ � = { a γ a − 1 } a ∈ Γ | γ � = e g on ˜ ◮ g on M Γ-invariant metric ˜ M � 1 g ( γ n y , y ) ℓ g ( � γ � ) = min dist ˜ g ( γ · x , x ) = lim n dist ˜ x ∈ ˜ n →∞ M ◮ Top entropy = volume entropy = Γ-orbit growth 1 1 h top = lim R log vol ˜ g ( B x , R ) = lim R log #(Γ · y ∩ B x , R ) R →∞ R →∞ ◮ Bowen-Margulis measure µ BM vs. Patterson-Sullivan current m PS Meas( SM ) φ t M ) φ t × Γ Meas( S ˜ Meas( ∂ � M × ∂ ˜ M ) Γ ↔ ↔ 4/13

Metrics on Negatively Curved Groups 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group ◮ D Γ = { left invariant metrics on Γ q.i. to a word metric } / ∼ 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group ◮ D Γ = { left invariant metrics on Γ q.i. to a word metric } / ∼ where d 1 ∼ d 2 if | d 1 − d 2 | is bounded 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group ◮ D Γ = { left invariant metrics on Γ q.i. to a word metric } / ∼ where d 1 ∼ d 2 if | d 1 − d 2 | is bounded Examples Γ = π 1 ( M , x ) with [ d g ] where d g , x ( γ 1 , γ 2 ) = dist ˜ g ( γ 1 . x , γ 2 . x ) 1 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group ◮ D Γ = { left invariant metrics on Γ q.i. to a word metric } / ∼ where d 1 ∼ d 2 if | d 1 − d 2 | is bounded Examples Γ = π 1 ( M , x ) with [ d g ] where d g , x ( γ 1 , γ 2 ) = dist ˜ g ( γ 1 . x , γ 2 . x ) 1 Note: d g , x ∼ d g , y because | d g , x − d g , y | ≤ dist ˜ g (Γ . x , Γ . y ) ≤ diam( M , g ). 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group ◮ D Γ = { left invariant metrics on Γ q.i. to a word metric } / ∼ where d 1 ∼ d 2 if | d 1 − d 2 | is bounded Examples Γ = π 1 ( M , x ) with [ d g ] where d g , x ( γ 1 , γ 2 ) = dist ˜ g ( γ 1 . x , γ 2 . x ) 1 Note: d g , x ∼ d g , y because | d g , x − d g , y | ≤ dist ˜ g (Γ . x , Γ . y ) ≤ diam( M , g ). Γ → Isom( X ) where X is CAT(-1) space, and Γ is convex cocompact 2 5/13

Metrics on Negatively Curved Groups General Setting ◮ Γ torsion free Gromov-hyperbolic group ◮ D Γ = { left invariant metrics on Γ q.i. to a word metric } / ∼ where d 1 ∼ d 2 if | d 1 − d 2 | is bounded Examples Γ = π 1 ( M , x ) with [ d g ] where d g , x ( γ 1 , γ 2 ) = dist ˜ g ( γ 1 . x , γ 2 . x ) 1 Note: d g , x ∼ d g , y because | d g , x − d g , y | ≤ dist ˜ g (Γ . x , Γ . y ) ≤ diam( M , g ). Γ → Isom( X ) where X is CAT(-1) space, and Γ is convex cocompact 2 Γ - Gromov hyperbolic, [ d ] where d - a word metric 3 5/13