Groups acting on Hyperbolic Spaces Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin New York City, 2013 Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Preliminaries I Definition Let Λ be an ordered abelian group. A Λ-metric on a set X is a function d : X × X → Λ such that: 1) d ( x , y ) ≥ 0 for any x , y and = 0 if and only if x = y . 2) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) for any x , y , z Definition If ( X , d ) is a Λ-metric space, the Gromov product with respect to a basepoint o is defined as c ( x , y ) = ( d ( x , o ) + d ( y , o ) − d ( x , y )) / 2. We say X is δ -hyperbolic if, for any x , y , z , c ( x , y ) ≥ min { c ( x , z ) , c ( y , z ) } − δ . This does not depend upon the choice of basepoint. In the specific case where Λ is Z n or R n , we will assume that δ = ( δ, 0 , ..., 0) since otherwise all subspaces of height smaller than δ would be unaffected by the hyperbolicity condition. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Preliminaries II Definition Let G be a group a Λ-length function on G is a function l : G → Λ such that: 1) l ( g ) = 0 if g = 1. 2) l ( g ) = l ( g − 1 ) for any g 3) l ( gh ) ≤ l ( g ) + l ( h ) for any g , h Define d l ( g , h ) = l ( g − 1 h ) and g ∼ l h if d l ( g , h ) = 0. It is easy to see that l is a length function if and only if ( G / ∼ l , d l ) is a metric space. Since G acts by isometries on G / ∼ l , we can see length function as equivalent to group actions. Definition A length function is δ -hyperbolic if ( G / ∼ l , d l ) is δ -hyperbolic. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Preliminaries III Definition A length function l is regular if there exists some k ∈ N such that, for any g , h , there exist c with l ( c ) + l ( c − 1 g ) < l ( g ) + k δ , l ( c ) + l ( c − 1 h ) < l ( h ) + k δ and l ( g − 1 c ) + l ( c − 1 h ) < l ( g − 1 h ) + 2 δ . Definition An action is regular if there exists a k ∈ N such that, for any g , h ∈ G there exists a c such that cx is in the k δ -neighborhood of the inner δ -triangle of { x , gx , hx } . This property doesn’t depend on x . Remark In general, an action is regular if and only if its associated length function is regular. In particular, all co-compact actions are regular. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Z n -hyperbolic groups Theorem (Grecianu and Miasnikov, 2013 - preprint) Let G be a finitely generated torsion-free group and l : G → Z n a regular δ -hyperbolic length function such that: 1. { g ∈ G : l ( g ) ≤ ( n , 0 , ..., 0) } is finite for any n 2. { g ∈ G : ht ( l ( g )) = 1 } is finitely generated. In that case, there exists an ascending chain G 1 < G 2 < ... < G n = G such that G 1 is a word hyperbolic group and, for any k, G k +1 is an HNN extension of G k with a finite number of stable letters and whose associated subgroups are virtually Z i with i ≤ k. If G acts properly and co-compactly on a Z n -metric space and the stabilizer of a Z -subspace is finitely generated, then we have that the above theorem applies. We will refer to such groups as Z n -hyperbolic by analogy. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Embeddings I Theorem (Grecianu, Kvaschuk, Miasnikov and Serbin, 2013) Let ( X , d ) be a δ -hyperbolic Z -space. Then there exists a δ ′ -hyperbolic graph Γ 1 ( X ) such that X embeds quasi-isometrically into Γ 1 ( X ) . δ ′ depends linearly on δ . For any ϕ an isometry of X there exists an isometry of Γ 1 ( X ) , ϕ , such that ϕ | X = ϕ and ∂ X = ∂ Γ 1 ( X ) . This results means that we can pass freely from group actions on hyperbolic graphs and length functions in Z . Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Embeddings II Theorem Let ( X , d ) be a δ -hyperbolic Z n -space which is simple and regular. Then there exists a (1 , 8 δ ′′ ) -quasi-geodesic δ ′′ -hyperbolic metric space � X which is geodesic on its Z -subspaces such that X embeds X. δ ′′ depends quadratically on δ . quasi-isometrically into � For any ϕ an isometry of X there exists an isometry of � X , � ϕ , such that ϕ | X = ϕ . � The condition of the original space being simple is a technical one, roughly equivalent to saying that all Z -subspaces have a point on the boundary which tends towards all other Z -subspaces. In practice, our examples will have this property. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Length functions and group actions Corollary Let G be a group acting regularly on a hyperbolic space X and a Z n -tree � G y = { 1 } . Then G Y such that there exist x ∈ X and y ∈ Y with G x acts regularly on a quasi-geodesic Z n +1 -hyperbolic space which is geodesic on its Z -subspaces. In general, the same is true if G acts regularly on a hyperbolic space X and on n trees Y n such that we can choose x ∈ X , y n ∈ Y n with � G y 1 � ... � G y n = { 1 } . G x � ... � G y n ) is finitely generated and its If we also have that G y (or G y 1 action on X is proper, we have that G is Z n +1 -hyperbolic. It is usually considerably easier to construct a length function in this way than to construct a Z n +1 -space and prove that G acts nicely on it. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Applications Theorem (Grecianu and Miasnikov, 2013 - preprint) Let G be a finite HNN extension of a torsion-free hyperbolic group whose associated subgroups are non-conjugate, primitive cyclic subgroups. Then G is Z 2 -hyperbolic. Theorem Let G be a group which is weakly hyperbolic relative to a free subgroup H. Suppose that H has a complement in G, K, such that G admits a presentation of the form: � H , K | h 1 k 1 h ′ 1 k ′ 1 , ... � where h i , h ′ i ∈ H, k i , k ′ i ∈ K and { h 1 , h ′ 1 , ... } is a basis for H. Then G is Z 2 -hyperbolic. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
Conjectures Conjecture Let G be a finite HNN extension of a Z n -hyperbolic group whose associated subgroups are non-conjugate, maximal free abelian groups of rank at most n, then G is Z n +1 -hyperbolic. Conjecture Let G be weakly hyperbolic relative to a subgroup H ≃ Z n ∗ F where F is free and H has a complement K such that G allows a presentation like the one before, then G is Z n +1 -hyperbolic. Conjecture If G is Z n -hyperbolic, then it allows a quasi-convex hierarchy. Andrei-Paul Grecianu, joint work with O.Kharlampovich, A. Kvaschuk, A. Miasnikov and D. Serbin Groups acting on Hyperbolic Spaces
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