Let G = Z and X = { 1 } . Let G = Z and X = { 1 } . Then ( G , X ) is - PowerPoint PPT Presentation

Q UASI - PARABOLIC STRUCTURES ON GROUPS Sahana Balasubramanya 1 University of M¨ unster V IRTUAL G EOMETRIC G ROUP T HEORY C ONFERENCE CIRM, J UNE 2020 1 Partly joint work with D.Osin, C.Abbott and A.Rasmussen

I NTRODUCTION G is a group

I NTRODUCTION G is a group ↓ Pick a generating set X (not necessarily finite)

I NTRODUCTION G is a group ↓ Pick a generating set X (not necessarily finite) ↓ Construct the Cayley graph Γ( G , X )

I NTRODUCTION G is a group ↓ Pick a generating set X (not necessarily finite) ↓ Construct the Cayley graph Γ( G , X ) ↓ G � Γ( G , X ) is isometric and cobounded

Let G = Z and X = {± 1 } .

Let G = Z and X = {± 1 } . Then Γ( G , X ) is

Let G = Z and X = {± 1 } . Then Γ( G , X ) is If X = G , then Γ( G , X ) is

Let G = Z and X = {± 1 } . Then Γ( G , X ) is If X = G , then Γ( G , X ) is

D EFINITION (C OMPARING GENERATING SETS ; ABO) Let X , Y be two generating sets of a group G . We say that X is dominated by Y , written X � Y , if sup | y | X < ∞ . y ∈ Y

D EFINITION (C OMPARING GENERATING SETS ; ABO) Let X , Y be two generating sets of a group G . We say that X is dominated by Y , written X � Y , if sup | y | X < ∞ . y ∈ Y � is a preorder on the set of generating sets of G and therefore it induces an equivalence relation by: X ∼ Y ⇔ X � Y and Y � X .

D EFINITION (C OMPARING GENERATING SETS ; ABO) Let X , Y be two generating sets of a group G . We say that X is dominated by Y , written X � Y , if sup | y | X < ∞ . y ∈ Y � is a preorder on the set of generating sets of G and therefore it induces an equivalence relation by: X ∼ Y ⇔ X � Y and Y � X . We denote the equivalence class of X by [ X ] .

◮ If X ⊂ Y , then Y � X (Inclusion reversing)

◮ If X ⊂ Y , then Y � X (Inclusion reversing) ◮ [ X ] � [ Y ] ⇐ ⇒ X � Y

◮ If X ⊂ Y , then Y � X (Inclusion reversing) ◮ [ X ] � [ Y ] ⇐ ⇒ X � Y ◮ If G has a finite generating set X , then [ X ] is the largest structure

◮ If X ⊂ Y , then Y � X (Inclusion reversing) ◮ [ X ] � [ Y ] ⇐ ⇒ X � Y ◮ If G has a finite generating set X , then [ X ] is the largest structure ◮ If [ X ] = [ Y ] , then Γ( G , X ) is quasi-isometric to Γ( G , Y )

T HE POSET OF HYPERBOLIC STRUCTRES D EFINITION (ABO) A hyperbolic structure on G is an equivalence class [ X ] such that Γ( G , X ) is hyperbolic.

T HE POSET OF HYPERBOLIC STRUCTRES D EFINITION (ABO) A hyperbolic structure on G is an equivalence class [ X ] such that Γ( G , X ) is hyperbolic. We denote the set of hyperbolic structures by H ( G ) and endow it with the order induced from above.

T HE POSET OF HYPERBOLIC STRUCTRES D EFINITION (ABO) A hyperbolic structure on G is an equivalence class [ X ] such that Γ( G , X ) is hyperbolic. We denote the set of hyperbolic structures by H ( G ) and endow it with the order induced from above. Elements of H ( G ) � Equivalence classes of cobounded actions of G on hyperbolic spaces (up to a natural equivalence)

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G )

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures.

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures.

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2 ◮ H qp ( G ) contains quasi-parabolic structures.

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2 ◮ H qp ( G ) contains quasi-parabolic structures. i.e. | Λ( G ) | = ∞ and G fixes a point of ∂ X

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2 ◮ H qp ( G ) contains quasi-parabolic structures. i.e. | Λ( G ) | = ∞ and G fixes a point of ∂ X ◮ H gt ( G ) contains general type structures. i.e.

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2 ◮ H qp ( G ) contains quasi-parabolic structures. i.e. | Λ( G ) | = ∞ and G fixes a point of ∂ X ◮ H gt ( G ) contains general type structures. i.e. | Λ( G ) | = ∞ and G has no fixed points on ∂ X .

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2 ◮ H qp ( G ) contains quasi-parabolic structures. i.e. | Λ( G ) | = ∞ and G fixes a point of ∂ X ◮ H gt ( G ) contains general type structures. i.e. | Λ( G ) | = ∞ and G has no fixed points on ∂ X . ◮ Parabolic actions are never cobounded

S OME THEOREMS AND MOTIVATION T HEOREM (ABO) For any group G, H ( G ) = H e ( G ) ⊔ H ℓ ( G ) ⊔ H qp ( G ) ⊔ H gt ( G ) ◮ Let Λ( G ) denote the limit points of G on ∂ X ◮ H e ( G ) contains elliptic structures. i.e. | Λ( G ) | = 0 H e ( G ) = { [ G ] } always and is the smallest structure ◮ H ℓ ( G ) contains lineal structures. i.e. | Λ( G ) | = 2 ◮ H qp ( G ) contains quasi-parabolic structures. i.e. | Λ( G ) | = ∞ and G fixes a point of ∂ X ◮ H gt ( G ) contains general type structures. i.e. | Λ( G ) | = ∞ and G has no fixed points on ∂ X . ◮ Parabolic actions are never cobounded ◮ H ( G ) is a way to study all possible cobounded actions of a group on hyperbolic spaces, upto q.i.

T HEOREM (ABO) For every n ∈ N , there exists a group G n such that |H ℓ ( G n ) | = n and |H qp ( G n ) | = |H gt ( G n ) | = 0 .

T HEOREM (ABO) For every n ∈ N , there exists a group G n such that |H ℓ ( G n ) | = n and |H qp ( G n ) | = |H gt ( G n ) | = 0 . T HEOREM (ABO) For every n ∈ N , there exists a group H n such that |H gt ( H n ) | = n and |H qp ( H n ) | = |H ℓ ( H n ) | = 0 .

T HEOREM (ABO) If [ A ] ∈ H qp ( G ) , then there exists [ B ] ∈ H ℓ ( G ) such that [ B ] � [ A ] .

T HEOREM (ABO) If [ A ] ∈ H qp ( G ) , then there exists [ B ] ∈ H ℓ ( G ) such that [ B ] � [ A ] . ◮ Consequence of the Buseman pseudocharacter (Manning)

T HEOREM (ABO) If [ A ] ∈ H qp ( G ) , then there exists [ B ] ∈ H ℓ ( G ) such that [ B ] � [ A ] . ◮ Consequence of the Buseman pseudocharacter (Manning) T HEOREM (ABO) H qp ( Z wr Z ) contains an antichain of cardinality continuum.

T HEOREM (ABO) If [ A ] ∈ H qp ( G ) , then there exists [ B ] ∈ H ℓ ( G ) such that [ B ] � [ A ] . ◮ Consequence of the Buseman pseudocharacter (Manning) T HEOREM (ABO) H qp ( Z wr Z ) contains an antichain of cardinality continuum. ◮ Obtained by factoring through Z n wr Z acting on the Bass-Serre tree.

Q UESTIONS 1. Does there exist a group such that |H qp ( G ) | is non-empty and finite ?