Groups acting on Hyperbolic Spaces Andrei-Paul Grecianu, joint work - - PowerPoint PPT Presentation

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 Zn or Rn, 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 dl(g, h) = l(g −1h) and g ∼l h if dl(g, h) = 0. It is easy to see that l is a length function if and only if (G/ ∼l, dl) 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, dl) 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−1g) < l(g) + kδ, l(c) + l(c−1h) < l(h) + kδ and l(g −1c) + l(c−1h) < l(g −1h) + 2δ. 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 kδ-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 → Zn 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 G1 < G2 < ... < Gn = G such that G1 is a word hyperbolic group and, for any k, Gk+1 is an HNN extension of Gk with a finite number of stable letters and whose associated subgroups are virtually Zi with i ≤ k. If G acts properly and co-compactly on a Zn-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 Zn-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 Γ1(X) such that X embeds quasi-isometrically into Γ1(X). δ′ depends linearly on δ. For any ϕ an isometry of X there exists an isometry of Γ1(X), ϕ, such that ϕ|X = ϕ and ∂X = ∂Γ1(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 Zn-space which is simple and regular. Then there exists a (1, 8δ′′)-quasi-geodesic δ′′-hyperbolic metric space X which is geodesic on its Z-subspaces such that X embeds quasi-isometrically into

• X. δ′′ depends quadratically on δ.

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 Zn-tree Y such that there exist x ∈ X and y ∈ Y with Gx Gy = {1}. Then G acts regularly on a quasi-geodesic Zn+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 Yn such that we can choose x ∈ X, yn ∈ Yn with Gx Gy1 ... Gyn = {1}. If we also have that Gy (or Gy1 ... Gyn) is finitely generated and its action on X is proper, we have that G is Zn+1-hyperbolic. It is usually considerably easier to construct a length function in this way than to construct a Zn+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 Z2-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|h1k1h′

1k′ 1, ...

where hi, h′

i ∈ H, ki, k′ i ∈ K and {h1, h′ 1, ...} is a basis for H.

Then G is Z2-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 Zn-hyperbolic group whose associated subgroups are non-conjugate, maximal free abelian groups of rank at most n, then G is Zn+1-hyperbolic. Conjecture Let G be weakly hyperbolic relative to a subgroup H ≃ Zn ∗ F where F is free and H has a complement K such that G allows a presentation like the one before, then G is Zn+1-hyperbolic. Conjecture If G is Zn-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