Warped cones, profinite completions, coarse embeddings and property - PowerPoint PPT Presentation

. Warped cones, profinite completions, coarse embeddings and property A . Damian Sawicki Institute of Mathematics Polish Academy of Sciences S´ eminaire Groupes et Analyse Universit´ e de Neuchˆ atel 21 October 2015 Partly based on joint work with Piotr W. Nowak. Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Warped metric on a Γ-space . Γ = � S � , | S | < ∞ Γ � ( X , d ) . Definition (Roe, 2005) . Warped metric d Γ is the largest metric satisfying: d Γ ( x , x ′ ) ≤ d ( x , x ′ ) , d Γ ( x , sx ) ≤ 1 ∀ s ∈ S . . Damian Sawicki Warped cones, coarse embeddings and property A

. Warped metric on a Γ-space . Γ = � S � , | S | < ∞ Γ � ( X , d ) . Definition (Roe, 2005) . Warped metric d Γ is the largest metric satisfying: d Γ ( x , x ′ ) ≤ d ( x , x ′ ) , d Γ ( x , sx ) ≤ 1 ∀ s ∈ S . . . Proof of correctness . 1. There exists a metric satisfying the two: min( d , 1). 2. Supremum of metrics is a metric: sup d ( x , z ) ≤ sup d ( x , y ) + d ( y , z ) ≤ sup d ( x , y ) + sup d ( y , z ) . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . . . . Warped metric on a Γ-space: geometric intuition . ( X , d ) – geodesic space For each pair of points ( x , sx ) glue an interval of length 1 between x and sx . Calculate the path metric in the space with all the extra intervals. Its restriction to X is the warped metric d Γ ! Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Warped metric on a Γ-space: geometric intuition . ( X , d ) – geodesic space For each pair of points ( x , sx ) glue an interval of length 1 between x and sx . Calculate the path metric in the space with all the extra intervals. Its restriction to X is the warped metric d Γ ! . Corollary . Being a quasi-geodesic space is preserved by warping. . Damian Sawicki Warped cones, coarse embeddings and property A

. Warped metric on a Γ-space: geometric intuition . ( X , d ) – geodesic space For each pair of points ( x , sx ) glue an interval of length 1 between x and sx . Calculate the path metric in the space with all the extra intervals. Its restriction to X is the warped metric d Γ ! . Corollary . Being a quasi-geodesic space is preserved by warping. . . Example (Hyun Jeong Kim, 2006) . X = R 2 , Γ = Z acts by rotating by angle θ . There are infinitely many non-quasi-isometric warped planes ( R 2 , d Z ) depending on θ . . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Warping: general motivation . d Γ ( x , sx ) ≤ 1 Damian Sawicki Warped cones, coarse embeddings and property A

. Warping: general motivation . d Γ ( x , sx ) ≤ 1 = ⇒ dist( id ( X , d Γ ) , γ ) ≤ | γ | = ⇒ Rich Roe algebras. . Conjecture (Drut ¸u–Nowak, 2015) . Warped cones over actions with a spectral gap violate the coarse Baum–Connes conjecture. . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Warped cones . Γ � Y – compact subset of S n ⊆ R n +1 X – infinite cone over Y with the induced Γ action: { ty | t ∈ [0 , ∞ ) , y ∈ Y } ⊆ ( R n +1 , d ) Damian Sawicki Warped cones, coarse embeddings and property A

. Damian Sawicki Warped cones, coarse embeddings and property A

. Damian Sawicki Warped cones, coarse embeddings and property A

. Damian Sawicki Warped cones, coarse embeddings and property A

. Damian Sawicki Warped cones, coarse embeddings and property A

. Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Warped cones . Γ � Y – compact subset of S n ⊆ R n +1 X – infinite cone over Y with the induced Γ action: { ty | t ∈ [0 , ∞ ) , y ∈ Y } ⊆ ( R n +1 , d ) Damian Sawicki Warped cones, coarse embeddings and property A

. Warped cones . Γ � Y – compact subset of S n ⊆ R n +1 X – infinite cone over Y with the induced Γ action: { ty | t ∈ [0 , ∞ ) , y ∈ Y } ⊆ ( R n +1 , d ) Notation: O Y . . = ( X , d ), O Γ Y . . = ( X , d Γ ). . Non-example: the case of a finite Γ . If Γ is finite, then O Γ Y ≃ O Y / Γ, e.g.: for the antipodal action Z 2 � S n : O Z 2 S n ≃ O R P n ; for rational θ : O Z S 1 = O Z k S 1 ≃ O S 1 / Z k ≃ O S 1 = R 2 . . Damian Sawicki Warped cones, coarse embeddings and property A

. Motivation / outline of the talk . Warped cones as a generalisation of box spaces: relation of equivariant properties of Γ and coarse properties of O Γ Y . Refinement of the former: relation of dynamic properties of Γ � Y and coarse properties of O Γ Y . Spaces with interesting coarse properties, e.g.: coarsely embeddable in ℓ p but without property A; not coarsely embeddable into any Banach space of non-trivial type. Damian Sawicki Warped cones, coarse embeddings and property A

. . . . . . . Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . . . . Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . Equivalently: for sequences ( x n ), ( x ′ n ) d ( x n , x ′ ⇒ d ( f ( x n ) , f ( x ′ n ) → ∞ ⇐ n )) → ∞ . . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . . . . Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . . . . Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . . . Γ is amenable if for each ε > 0 and finite R ⊆ Γ there exists µ ∈ S ( ℓ 1 (Γ)) such that: � µ − s µ � ≤ ε if s ∈ R ; supp µ is finite. . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . . . Γ is amenable if for each ε > 0 and finite R ⊆ Γ there exists µ ∈ S ( ℓ 1 (Γ)) such that: � µ − s µ � ≤ ε if s ∈ R ; supp µ is finite. . . Definition . ( X , d ) has property A if for each ε > 0 and R < ∞ there is a map X ∋ x �→ A x ∈ S ( ℓ 1 ( X )) and a constant S < ∞ such that: � A x − A y � ≤ ε if d ( x , y ) ≤ R ; supp A x ⊆ B ( x , S ). . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . . . Definition . ( X , d ) has property A if for each ε > 0 and R < ∞ there is a map X ∋ x �→ A x ∈ S ( ℓ 1 ( X )) and a constant S < ∞ such that: � A x − A y � ≤ ε if d ( x , y ) ≤ R ; supp A x ⊆ B ( x , S ). . Damian Sawicki Warped cones, coarse embeddings and property A

. Recall the definitions . . Definition . f : X → Y is a coarse embedding if there are non-decreasing functions ρ − , ρ + : R + → R + with lim r →∞ ρ ± ( r ) = ∞ such that ρ − ( d ( x , x ′ )) ≤ d ( f ( x ) , f ( x ′ )) ≤ ρ + ( d ( x , x ′ )) . . . Definition . ( X , d ) has property A if for each ε > 0 and R < ∞ there is a map X ∋ x �→ A x ∈ S ( ℓ 1 ( X )) and a constant S < ∞ such that: � A x − A y � ≤ ε if d ( x , y ) ≤ R ; supp A x ⊆ B ( x , S ). . . Implications . Amenability = ⇒ property A = ⇒ coarse embedding in ℓ 1 or ℓ 2 . . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . . . . Box space . Γ = Γ 0 > Γ 1 > Γ 2 > . . . – residual chain G n = Cay(Γ / Γ n , S ) Box space � G n is the sequence ( G n ) or, more concretely, ∐ n G n with any metric such that dist( G n , G m ) → ∞ as max( n , m ) → ∞ . Damian Sawicki Warped cones, coarse embeddings and property A

. . . . Box space . Γ = Γ 0 > Γ 1 > Γ 2 > . . . – residual chain G n = Cay(Γ / Γ n , S ) Box space � G n is the sequence ( G n ) or, more concretely, ∐ n G n with any metric such that dist( G n , G m ) → ∞ as max( n , m ) → ∞ . . Theorem (Guentner–Roe, 2003) . � G n has property A ⇐ ⇒ Γ is amenable. � G n embeds coarsely into a Hilbert space = ⇒ Γ has the Haagerup property. . Damian Sawicki Warped cones, coarse embeddings and property A