[Paper Review] Uniform embeddability of relatively hyperbolic groups

Marius Dădărlat, Erik Guentner|ArXiv.org|Jan 27, 2005

Geometric and Algebraic Topology10 references4 citations

TL;DR

This paper establishes that a finitely generated group hyperbolic relative to a finite family of subgroups is uniformly embeddable in a Hilbert space if and only if each of those subgroups is uniformly embeddable. The authors introduce a novel gluing technique for uniform embeddability using partitions of unity and equi-uniform embeddability of subspaces, which they apply to prove the permanence property for relatively hyperbolic groups.

ABSTRACT

Let $Γ$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $Γ$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly embeddable in a Hilbert space.

Motivation & Objective

• To establish a permanence property for uniform embeddability in the context of relatively hyperbolic groups.
• To develop a general gluing technique for uniform embeddability of metric spaces based on partitions of unity and equi-uniform embeddability of subspaces.
• To provide a conceptual framework that unifies and extends known permanence results for uniformly embeddable spaces and C*-exact groups.
• To recover and re-derive Ozawa's result on C*-exactness of relatively hyperbolic groups using the same methodological approach.
• To bridge the gap between geometric group theory and functional analytic properties like uniform embeddability and Property A.

Proposed method

• Introduce a gluing technique using a partition of unity $(\varphi_i)_{i \in I}$ on a metric space $X$ such that the associated map $\Phi: X \to \ell^1(I)$ is Lipschitz.
• Require that the supports $\mathrm{supp}(\varphi_i)$ are equi-uniformly embeddable, meaning they admit a uniform family of embeddings into Hilbert space with controlled distortion.
• Use the uniform control of the partition and the equi-uniform embeddability of the pieces to construct a uniform embedding of the whole space $X$ into a Hilbert space.
• Apply the gluing technique to the metric space $X = (\Gamma, d_S)$, where $\Gamma$ is relatively hyperbolic, and $Y = (\Gamma, d_{S \cup \mathcal{H}})$ with $p: X \to Y$ the identity map.
• Leverage Osin’s recursive decomposition of balls $B(n)$ in $\Gamma$ to show that each $B(n) = p^{-1}(B_Y(e,n))$ is uniformly embeddable if the peripheral subgroups $H_k$ are.
• Apply Corollary 4.7 to conclude that $X = \Gamma$ is uniformly embeddable if all $H_k$ are uniformly embeddable.

Experimental results

Research questions

• RQ1 Under what conditions does the uniform embeddability of subspaces imply the uniform embeddability of the entire metric space?
• RQ2 Can a general gluing technique be developed to unify permanence results for uniform embeddability and Property A in metric spaces?
• RQ3 Does the uniform embeddability of a relatively hyperbolic group depend solely on the uniform embeddability of its peripheral subgroups?
• RQ4 How does the gluing method compare to existing techniques in finite asymptotic dimension and Property A theory?
• RQ5 Can the same method be used to recover known results on C*-exactness of relatively hyperbolic groups?

Key findings

• A finitely generated group $\Gamma$ that is hyperbolic relative to a finite family of subgroups $\{H_1, \dots, H_n\}$ is uniformly embeddable in a Hilbert space if and only if each $H_i$ is uniformly embeddable in a Hilbert space.
• The proof relies on a new gluing technique using partitions of unity and equi-uniform embeddability of subspaces, which allows the uniform embedding of the whole space from embeddings of its parts.
• The method applies to the inverse image of balls under the identity map $p: (\Gamma, d_S) \to (\Gamma, d_{S \cup \mathcal{H}})$, showing that each $B(n)$ is uniformly embeddable when the $H_k$ are.
• The result recovers Ozawa’s theorem on C*-exactness of relatively hyperbolic groups, demonstrating the method’s broader applicability.
• The approach provides a conceptual and unified treatment of permanence properties for uniformly embeddable spaces and C*-exact groups.
• The technique is versatile and extends beyond the scope of this paper, with potential for further applications in geometric group theory and coarse geometry.